Good job! Thanks for the update. The idle was probably set to compensate for all the problems you found. When you solved the problems, the idle setting was now high.

Good day all,

This will hopefully be the last update on this topic. 
Finally got around to doing the head gasket and, sure enough, that was the failure point which was causing the loud noises I was experiencing. Attached is an image of the head gasket and the failure can be clearly seen as there is no gasket material left to front of cylinder 1. I found some other things as I was tearing the engine down like the valves being very tight and not at the right clearance and some of the rockers not being seated properly in the cups of the push rods. I was able to address all of these things on reassembly. 
Upon replacing the gasket, doing a valve adjustment, and putting it all back together, the mini fired right up and sounded excellent. No odd noises, no odd smoke out the rear or engine area, or other abnormalities. 
The last things I need to finish up is to burp the cooling system and adjust the idle as the idle speed was really high at around 1500 rpm for some odd reason. Afterwards I'll take it out on short drives to see if all is well and in about 100 miles I'll recheck all of the studs and whatnot to ensure they are still at the right torque and recheck the valves to ensure they're still at the desired 0.012-in spacing. 
Thank you everyone for your insight and help!

My suspicion agrees with your suspicion. I strongly recommend you change the by-pass hose to a known good brand of heater hose. The bellows type was made to do quick/easy replacements without removing the head. But they are know to not last long.

Thank you for the suggestion Dan!

I'm pretty confident that I have a failed head gasket as the noise problem I've been experiencing, before I started taking things apart, sounded like it was coming from the cylinder 1 region. Maybe my desperation is getting the better of me but, when I ran the engine with it partly disassembled, I'm pretty sure I also saw exhaust fumes coming out the side of the block where the holes are in the head gasket. I ordered the gasket kit with the Payen black composite head gasket which should arrive by Monday so I plan to start the job next week.
As I am working on replacing the head gasket, I will make sure to inspect and swap(if necessary) the bypass hose. 
I will provide another update once I get the head off with pictures of the head and the head gasket to see if my theory on the failing head gasket is correct. If it isn't I may go bang my head against the nearest wall...

Although I don't doubt your suspicion, before getting into replacing the head gasket, i would replace that by-pass hose - the little accordion hose going from the head to the water pump.

Those accordion hoses are notorious for failing. In your picture, neither hose clamp looks straight, especially the upper one. There also appears to be coolant stain above the head gasket, possibly coming down from the top of the by-pass.

A better solution is a straight piece of heater hose - sorry I don't recall the right size (maybe 1/2"??). All you need is about 2", cut to length to fit fully from the head to the water pump, and 2 new worm-type hose clamps, stainless steel if you can find them.
Once you get the old hose out, clean the two spigots. Open the hose clamps as far as they will go without coming apart. Place them over the spigots. Take the hose, bend it in two with a pair of vice-grip pliers. Push the hose in so its ends touch the spigots. Release the pliers so that the hose ends push against the spigots. With a little massaging, the hose will pop into place. Then tighten the hose clamps. Holding the hose clamps over the spigots may be to awkward - if it doesn't work for you, you can open them up and slip them around the hose once it is in.

It should be relatively easier for you with the fan out of the way, though I've done it with the fan in.

Good evening all,

Another update. I was working on replacing the fan and what not when I decided to more thoroughly investigate the block to see if I can spot the location of the leak which started this whole debacle. During the investigation I briefly ran the engine to try and locate the source with the fan/radiator off and I believe I may have found the source of the leak! Unfortunately it looks to be a failing head gasket. I couldn't see it before with the fan on as the failure in the head gasket looks to be near cylinder 1, under the thermostat housing. There look to be 2 spots where the gasket has failed.
So, time for a head gasket replacement...
If anyone has any advice, useful tips, or resources/guides they can share, I would greatly appreciate it!

Cheers!

cut the fan belt and run the engine. If the noise disappears, it's tied to the belt. Further investigation needed. Fan our of balance?, alternator bearing noise.?   It does sound like popping from the engine.  I'd start with a valve adjustment before I remove the fan belt. And when you take the belt off, I'm assuming you were thinking of replacing the fan.

Hi all,

Thank you for your offer Donnie but I've already bought and installed a new manifold I bought from MiniSport, yet that didn't resolve my noise problem. If I happen to need another manifold for any reason, I will be in touch. Thanks again!
A new fan should be arriving tomorrow so I will be swapping that in and hopefully that is the source of my noise problem.
I bought the Mini from a small dealer in the US earlier this year and I believe it was imported a couple years ago. It's unfortunate that the noise developed after such a short time of my ownership but at least I am handy enough to be able to, hopefully, fix it myself and I do enjoy tinkering with cars.

Cheers and thank you all for sharing your knowledge and input!

fzxtreme.. I have couple spare exhaust manifold.. glad to send ya one for FREE. if you pay the postage.. btw. where did you  buy your mini from ?/ in the USA or the UK..??

The smooth face of the hub goes on the water pump.

+1 to what Ian suggests.
In your picture, you can see the hub of the fan is well within the shroud and the blades bend back toward the engine. Being plastic, these fans can distort over time. If it is that close to the breather, it may rub during higher engine rpm and may eventually snap a blade off. You don't want loose blades hitting your radiator.

When you put the new fan on, check which way round it goes - the blades are an airfoil shape and are about twice as effective with the thicker part of the airfoil toward the engine.

buy a new fan. they are cheap.

Good day all, thought I'd give an update as it's been a while.

So, I first tried just changing the manifold gasket to see if the noise went away and it didn't. Next, since the the exhaust manifold was quite rusty like Dan stated and maybe it developed a hole, I went ahead and replaced the exhaust manifold this past weekend but the noise remains!
When I had the carb/manifolds off I did look around extensively in the engine bay to see if there may be another region the noise could be coming from or if anything else looked odd. I didn't find any holes and the face of the exhaust ports didn't look damaged or distorted, which would cause a poor mating surface. Figured if a hole or another area had a leak, there would be clear evidence of it.
I did see that the radiator fan blades are quite close to the breather canister on the timing chain cover. I attached a picture of a top side view of the blades and canister where it shows that the blades protrude past the cowling.
From reading other threads on the matter, it seems to me that the fan should be fully enclosed in the cowling and not protrude past the cowling or it could contact the breather canister. I'm guessing that the fan itself is missing the necessary spacer or the spacer isn't the right size. 
With the car running it did look like the fan blades were contacting the canister and the noise I'm experiencing looks to originating from that area as well. I couldn't find any videos or sound clips of what a clashing fan blade sounds like but could the sound I'm experiencing be the sound of the fan blades colliding with the canister?
That is my next theory, as I've run out of other ideas, and I will work to better place the fan blades in the cowling and see what happens.

Looks like (and is in about the right place) for the standard fuel feed line. Flex line usually connects this to the carb.

Maybe while you have it on stands you can trace where the line goes? Probably heads back towards the tank.

Cheers, Ian

Thank you Ian and Dan for your input!
I got the car on stands to see if I could spot anything from underneath and there may be some black soot near the manifold so that supports the theory of a failing gasket. I tried to feel around the manifold to see if I can feel it but couldn't. I've ordered a new gasket and will begin the replacement soon. Thanks again for your input!
On a unrelated note, I spotted a disconnected hardline from what looks like it could be a vacuum line or some sort of return line near the radiator. Would anyone know what what it's for or what it connects to?
https://imgur.com/ngeskO0
https://imgur.com/rCr2DGC

+1 to Ian's diagnosis. Your exhaust manifold does look rusty. Perhaps it is beginning to fail. The gradually increasing noise suggests to me a hole that is getting bigger. The "popping" noise sounds to me like the exhaust pulses from the closest cylinder to the hole. Normally this kind of sound is attenuated within the exhaust system.

If you can't reach under/around the exhaust manifold while it is hot, try using a listening tube (hose or metal pipe) to locate the noise more closely.

Certainly sounds like an exhaust leak.... The video doesn’t help that much... you need to concentrate on the back of the engine...... from below would be good.

Before pulling anything apart I would be getting the car as high as possible (with the engine off :) and looking for either shiny bits (something rubbing - unlikely but possible) or black smudges where the exhaust gasses are escaping.

I would concentrate on the bottom of the manifold gasket or a bit lower down where the manifold and exhaust pipe join. To me it sounds like only a single cylinder which *might* implicate the head to manifold gasket on No 1 cylinder. You might be able to reach ...VERY!!!!! carefully under the manifold and actually feel the puffs of exhaust escaping....

Cheers, Ian

Good day all,

I recently acquired a 93 Mini 1300 with a carb and have been enjoying it on the weekends but earlier today the car may have developed an exhaust leak or maybe something more serious is brewing. To me it sounds like an exhaust leak but I wanted to ask others who are more experienced with these vehicles. 
I was on my way to cars and coffee and the car was driving normally up until it the engine started to get louder and louder. I pulled over to inspect it and found nothing obvious and decided to take the Mini home instead and investigate the noise further. It sounds like a popping noise, it sounds to be coming from the right side of the engine near the radiator (which is why I focus on that area in the video, linked below), and it doesn't seem to be coming from the underside of the engine. Driving it back home, the car didn't lose power or drive any differently once the noise developed and the noise does get louder as the engine revs higher.  When I would left off the throttle the noise would subside and the engine would quiet down to a normal level. Only when I would accelerate again or the car was stationary and idling would the noise return. This leads me to believe it has something to do with the exhaust side of things since the noise seems to occur when under idle/acceleration and not when engine braking/coasting. So my current plan is to replace the exhaust manifold gasket and further inspect the area as I take stuff off but I wanted to see if anyone may have some knowledge/experience they can bestow on me.

Here's a link to a short video of the engine noise if anyone may have any advice. I may post a driving video as well for further documentation.
https://www.youtube.com/shorts/RfkDXWHNbJc

Thanks for reading!