You said there were no leaks and that the pedal goes to the floor.  If it were the wheel cylinders you would have fluid dripping from somewhere.  Try rapidly pumping the brake pedal to see if it becomes firm and "higher".  Let us know what you find.

What did you do to/with the brake light switch?

As Bill said, old fuel can turn to gum and varnish if left for a few months.  However, make sure the valves are adjusted and ignition system is in order (tuned up) before you touch the carb.  You can fill the dashpot if you want... that won't affect the mixture.  Filling up the dashpot means removing the black screw cap and looking down inside.  You will see a steel tube in the center.  Fill the inner tube to within 1/4" of its top.  Any more than that will get sucked into the intake and burned.  Don't worry about "what oil" for now.  Put whatever engine oil you are using in there for now.  You can try other oils later once you are familiar with the carb.

Brake pedal going to floor is master cylinder problem in my book

Do you have vacuum boost front disc brakes?

Did you fill up with bad gas near Grapevine when the engine problems began? 

jcav.. sounds bit like fuel problem?/ how long had the mini been stored?? as  most folks use petrol with 10% ethanol..  which turns to glue in about six months in the gas tank.. and causes all kind of carb problems..  on the brakes.. kind sounds like your brake spliter is going back..  has two metal line on top and two coming out..  and it  mounted on bulkhead..  ( this are NOT cheap..about $275 ..) I had to replace all brake spliter on the Chokes.. ( go figure?? later bc

If it has a SU carb, you should start by checking the oil level in the carb dashpot. Remove the black cap and pull out the damper. There should be oil in the damper core - fill only the smallest diameter bore you see in there.

This is my first time with british cars and having a carb. I have had fuel injected aircooled VW's before this and i know their signs and symptoms well. The mini engine is new to me especially the carb. 

I figured it might be the rear brake cylinders. One evening the brakes were fine, and i was figuring out an issue with the tail lights. I pressed the brake pedal fairly hard a couple of times after tinkering with the brake light switch and the next day there wasn't any brake pressure.

Also, i picked up the mini in San Francisco and drove it to LA. It was around the grapevine (driving it fairly hard up it) that i developed the engine stumble or delay or the fuel delivery i'm not sure. It would drive fine till the car warmed up and then it would do it stumble or misfire or act like it wasn't getting enough fuel.

Im going to take a look at all fo the things you guys suggested and see what I can come up with.

If you pump the brake pedal does it get firmer and stop higher?  If the brake pedal does "pump up", does it stay up or will it sink slowly to the floor while you hold pressure on it?

It sounds like you are new to working on cars.  I caution you that there is more to switching to DOT-5 than simply flushing the system and pouring it in.  If you are not using DOT-5 now do not change at this time.  Resolve the problem immediately before you without throwing more variables in the mix.

EDIT:  Sorry, I neglected to comment on the engine issue.  As stated above, go through a complete tune up before looking for deep underlying problems.  Adjust the valves with the engine cold, then check and address the ignition system.  Always leave fuel adjustments for last.  

With the brakes, You probably need two rear brake cylinders and a master cylinder rebuild kit or a new master cylinder. Use DOT 5 when rebuilding, of course flush out the old brake fluid first.

Rear brakes may be in need of adjustment.
System may have air in the lines.
Master cylinder.

Carb adjustment/fuel and air mixture/needle adjustment

Timing
Ground
plug wires on correctly?

Someone may be relatively near and see your post here....have beer on hand  

I have a 1986 Austin Mini here in Albuquerque. I didn't see any brake fluid leaking anywhere and the master cylinder looked full, but the brake pedal is able to be pushed all the way down. It has no pressure. I'm not sure what to look for or what to do. 

The next issue is the motor on the highway or in city driving with the gas pedal pushed halfway down or all the way down stubles and feels like its miss firing or that the power is being cut out. I'm also sure not what to do here.