I compiled my own custom kernel in a 64 bit linux mint 16 installation. I've also heard that the 64 bit kernel could also run 32 bit applications too. Would it be possible to install it and use it in the 32-bit Ubuntu 13.10 installation or do I have to recompile it again for the 32-bit? If 32 bit Ubuntu would run on this kernel, Is there any problems I am going to encounter while using this kernel?
The following combinations will work:
In general running 32 bit software with a 64 bit kernel means having most of your system 64 bit and just running a few 32 bit applications - usually things like skype or steam for which we do not have source code and hence cannot recompile for a new processor.
In theory it should be possible to run an entirely 32-bit userland with only a 64 bit kernel. See, eg reports of people doing this successfully on Arch. But the package manager is unlikely to be happy with this, so it would involve quite a lot of DIY.
In the past this was an attractive option because running 32 bit applications in a 64 bit userland was fraught with problems. This is now largely solved, so running a mixed system in this way seems to have few advantages, and many disadvantages, compared to just running a pure 64 bit system.
I don't believe that you can.
Kernels exist for 32-bit and 64-bit systems, make sure that you have the kernel matching your system. If you have 64-bit, then install and use the 64bit kernel. If you have a 32-bit system, then use 32bit kernel.