Microsoft Office 2013 keeps crashing, need help

Uninstall Office, run SFC /SCANNOW in an admin command prompt. Allow to complete, restart, Install Office and patch. Does it work?

If no, create a new, test Windows User Profile (Account). Log into the new Windows Account and try again.

Office 2013 works just fine on Windows 7 64-bit.

Although a 32 bit version of office will work on a 64 bit operating system, is there any specific reason you decided to install the 32 bit version. If not why dont you install the 64 version of office. We have for several years used office 2013 on windows 7 machines without any problems.  This is for over a thousand users (all systems are imaged the same).

64-bit Office is generally not recommended because it does not coexist well with other 32-bit software (Adobe, Flash, Java and so on).

We have many dozens of client 64-bit computers all humming along fine with 32-bit Office. It remains the recommended approach by Microsoft

Start a command prompt, and type in "Outlook /safe" see if this loads without crashing.
If yes, disable plugins from startup.

For the rest Johns suggestion for a SFC /Scannow is a good way to go.
To add to that comment, there are further steps required once corruption has been detected. Running Sysnative's SFCFix.exe is an easy way to help on that area. See article on this tool/method here

instead of upgrading the 2010 version, why don't you uninstall that first - then install the 2013 fresh?

when trying to uninstall both office 2010 and office 2013 the uninstall wizard would pop up, then act like it was going to start, and then close out of itself.

Any suggestions?

I see your problem. Do not use two versions of Office on the same machine. Uninstall Office 2010 and all bits, uninstall Office 2013 and all bits, shut down, start up and install just Office 2013. That will fix your issue.

if it does not uninstall, try REVO to uninstall them :  http://www.revouninstaller.com/

If you are having issues try either as suggested by Nobus, or use the official Microsoft Uninstall tools located here.

Microsoft Office 2010 Uninstall tool
Microsoft Office 2013 Uninstall Tool

The Uninstall 2010 and 2013, then reinstall 2013 stopped the crashing and everything works fine now except for one of our companies Access databases. The database was created in Access 2007, which should still work in 2013, but whenever I try to enter the database error messages pop up. The error messages look like strings of code. Out of the 3 computers that I've tested it on, the one that works fine is the one I upgraded from 2010 to 2013.

Try (as a test) upgrading the Access DB to 2010 on another machine and then to 2013 on your machine to see if that works.

Through trial and error, and help from coworkers we found the solution