No idea what version are 64 bit compatible. What does the Mysql site have?
No problems with running 32 bit applications on Windows 7. In fact it may be better from a compatibilty perspective if you give it to others until everybody is using 64 bit OSes.
That said, there's nothing stopping you from building both 64 and 32 bit versions with the same code... but given your current problems, I'm sure you'd want to fix them first before even contemplating that.
There are different dll versions for the various clients. The DLL version MUST match the TMySQL* connector you use in the code. Welcome to MySQL hell...
Also, use 32 bit mysql client dlls for 32 bit applications, 64 bit for 64... but you probably know that by now.
Perhaps the mysql driver dll needs additional dlls as well (e.g. for compression and encryption) and will barf if they're not there.
Finally, you can try using tools like Sysinternals Process Monitor to determine what mysql dlls your application is trying to load.