Ah...teachers. I have a daughter who is a teacher, but in a classroom and she too does a lot with email so I can imagine being online teachers how much space this would take. I would recommend that you try to keep items like this on an external drive if possible, because your school items most definitely SHOULDN'T be lost.

Honestly, if possible I would recommend using a webmail for email if you can. Though if that is not possible if you are required to use an email address set up by your school, one thing you can do is use GMAIL accounts for email storage. Gmail allows a LOT of space for storage of email, around 7GB I believe.
The way to do this is open the GMAIL accounts, receive the mail in the regular accounts via Outlook but then forward the mail to the GMAIL account for online storage and delete it from Outlook. This way you could access it anytime you need to do so but it would help keep the mail OFF the computer for the school year but still safely stored and easily accessible.
If you need to correspond with a student concerning a specific mail or lesson just send that mail back to yourself at your "regular" school address that you use with Outlook. Send the student mail using Outlook and it would go to them via your "regular" mail address.

You might also consider an online storage site, https://www.dropbox.com/ as a place to store documents for free. Believe the site offers free 2GB of storage so you can have files and documents available on the site which you could access from any of your computers, from anywhere, when you save the document or file in a folder for dropbox. This one gets very good reviews. So you might check that out also.

If each of you opens a GMAIL account and a dropbox account that would give you each 9GB of online storage or roughly 18GB of space not being used on your computer. Both GMAIL and dropbox accounts are accessable from any computer, anywhere, as long as you have your login names and passwords.

I didn't really see anything in Add/Remove that looked out of the ordinary or extremely large. So I don't believe that is a problem either.

The slowness you are still noticing could also have to do with the need to defrag the computer, have you done this lately?