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There are many sources. There is a book on Corvette Water Leaks and Wind Noise solutions (thats also the name) which gives some procedures to find leaks. www.corvettefaq.com is also a good source of info.
Most likely sources are seams that have split (glue/sealant dried up), dried or cracked grommets or missing seals in the firewall.
First you need to remove your sill plates so you can remove your kick panels. Peel back the carpet off the firewall. Get someone to shine a torch from the engine bay side on the firewall. See any light inside????? If so patch/seal etc
Next step you do the same procedure with a gardren hose. Spray a LOW pressure stream of water over the firewall from the outside. See any leaks inside?
Feel up inside the side vents on the inside of the car(behind the kick panels) and see if it leaks there. It often cracks there as that is the seam between the fenders and the firewall. It is a long join that was glued at the factory
New grommets are available from almost any corvette vendor but if in a pinch a $2 tube of silicon sealer from Home Depot will also work.
You can fix it very cheap. Finding it is the first battle and usually the hardest part.
I solved my problem by sealing the gaps betwen the trim and the frame at each upper corner of the windshield. The water was running down the windshield pillars and under the dash...makes it seem like a firewall leak as you describe.
Recently I drove through a DOWNPOUR and only got a few drops in through the t-top seals. (Normally, I don't drive the car in heavy rain...but we got caught unexpectedly. The car handles quite well, even where there are several inches of water on the road...)
Welcome to the wonderful world of C3 Corvette ownership...I'll bet the remote start has fallen waaaaaaay down your to-do list
the first thing is to trickle a little water in the windshield wiper well area and see if that area is the problem. if so, dig out all the old caulk and reseal it with new caulk. hopefully this area is not rusted out because it's very $$$$ to fix. in regards to your door not coming unlocked, once you get the door open, put one hand on the bottom and one on the top and try to push it up & down. feel for the up & down movement and look at the hinge pin area while doing this, if you've got movement, most chevy dealers carry the pins and bushings needed to take the slop out of the hinges. if the hinge pins are worn the door "sags" making the door mechanism and lock assembly get in a bind , therefore preventing unlocking. on my 79 the drivers door hinges were worn more than the passenger door, but i replaced both sides, and my locks work much better, also remove the door panels and clean the door lock actuators,the electrical plugs, cylinder shafts, relube with lite grease, cost was about $20 and my time. good luck.
I tried undercoating and it didn't work...so I scraped it all out and cleaned up the area; then just laid down some glass fiber in the deep areas and poured some resin in there so that the it would seal, find the natural low levels, and sealed the well area.