First and foremost let's get this out of the way. Your hardware is based on x86 architecture. Here is some quick knowledge on that:
http://en.wikipedia.org/wiki/X86. The key takeaway there is that starting with the AMD Opteron the x86 registers were extended from 32-bit to 64-bit.
Now to go down your list:
1) Yes you can install 32-bit versions of an operating system. This is because you are really installing an x86 version of the OS. The OS is compiled for either the 32-bit or 64-bit register. Now if your hardware predates the extension to 64-bit, then you will only be able to install 32-bit. To answer your question more directly: It depends on what you need to do with your server. 64-bit allows for addressing > 3GB of RAM and is the future of the MS Server Operating Systems (2008 R2 is only 64-bit). Can whatever role or application you are installing support a 64-bit architecture?
2) Yes, through what Microsoft calls WOW64 (Windows 32-bit On Windows 64-bit). Now that does not mean that every piece of 32-bit software will run on a 64-bit Operating System, and it is best to check your software documentation to reference its support. For more on WOW64:
http://en.wikipedia.org/wiki/WOW643) Yes.
4) Yes
In closing, remember that 32 vs. 64 bit can raise a lot of questions, and if you are planning to make the jump consult all your installed software, drivers, peripherals, etc... before making a switch.