Q: When I transfer all the information from my old (2008) Macbook Pro to my 2013 Macbook Pro, does it transfer Everything?

I can tell you for sure that it will wipe EVERYTHING including the 3rd party software installed. If it came with Office Teacher Student it will be gone unless you have the disc to re-install. That really isn't a problem you can download Open Office and it's FREE.

Hopefully you used Migration Assistant to move from your old to your new Mac.  If you did, you will get everything you need.

Regarding Microsoft Office, I strongly recommend using Office 2011.  After all, you will be using a newer Operating System, so you should use the latest Office to keep up with the Operating System changes.  Regardless of which version you decide to use, once you first open it (to use it), you will be asked for the Access Code (that is on Office's box, or CD).

I also suggest you make sure to do software updates, both for Apple software (from the App Store, click on the "Updates" tab), and for Microsoft Office (once you open one of the programs and enter the access code, do Help > Check for Updates).

Office for Mac is third party ... absolutely.

Yes it is 3rd party.

You cannot legally do that.  You own the license for Office and it cannot be transferred.