Will compiling a compiler like GCC result in better performance?
A program compiled specifically to the target platform it is used on will usually perform better than a program compiled for a generic platform. Why is this? Knowledge about the harware can help the compiler align data to be cache friendly and choose an instruction ordering that plays well with a CPUs pipelining.
The most benefit is usally achieved by leveraging specific instruction sets such as SSE (in its various versions).
On the other hand, you should ask yourself if a programm like GCC is really CPU bound (much more likely it will be IO bound) and tuning its CPU performance provides any measurable benefit.
Will the code compiled by my own compiled compiler perform better
Hopefully not! Allowing a compiler to optimize a program should never change its behavior. No matter how you compiled your GCC, it should compile code to the same binaries as a generic binary distribution of GCC would.
If code compiled to the specific platform is faster than code compil for a generic platform, why dont we all ship code instead of binaries? Guess what, some linux distros actually follow this phillosophy, such as Gentoo. And while you're at it, make sure to built statically linked binaries, disk space is so cheap nowadays and it gives you at least another 0.001% of performance.
Alright, that was a bit sarcastic. The reason people distribute generic binaries is pretty obvious: It's geneirc, the lowest common denominator and it will work everywhere. Thats a big bonus in terms of flexibility and user friendlyness. I remember once compiling Gnome for my Gentoo box, it took a day or two! (But it must have been so much faster ;-) )
On the other hand, there are occassions where you want to get the best performance possible and it makes sense to build and optimize for specific architctures.