If I purchased some programs that advertise they work with AutoCAD 2013, can I expect them to work in 2015?

Possibly but there is no guarantee.  When in doubt....ask.  What programs are you looking at?  Have you visited the website of the company that makes the software?

I was looking at some add-ons here: 

http://partnerproducts.autodesk.com/catalog/category.aspSpecifically, the Weld Symbol program to start with, but it does not mention 2015.I haven't asked the company directly. Just started looking at this yesterday and thoughtsomebody might know if 2013, 2014 and 2015 are close enough that no rewriting would be necessary.I am not much of a programmer/developer. Just a user. I haven't played with Lisp in over 10 years. 

I'd imagine a symbol program would be fully compatible.