Upgrading Bicycle Gearing
Not brifters, but you can take your downtube shifters and use retroshift levers to get combined brake and and shifting. Essentially, they bolt a downtube shifter onto the hoods. Pretty cheap and reviews are generally positive. 

As for fitting more gears, you may be able to get a 7 speed freewheel, but you may better off taking your 6 speed freewheel to the LBS and swapping it one with better spacing between the cogs. 

Tires in 27" aren't too limited (Conti makes gatorskins in 27 inch, panaracer has some, etc.). But if you need to move to 700C, you need to make sure you can move your brakes down by 4 ish due to the 700c diameter change. The selection of new 27" wheels is more limited though (Sun CR-18, etc.), though can be found.