At work I pay into an "involuntary 401(a)" retirement plan. I'd like to also open up a Roth IRA on my own. Do the contributions I make through my work plan count against the $5000/year limit on the Roth?
No, your 401(a) contributions will not count against your Roth IRA limit. Contributing to other IRA accounts can reduce how much you can contribute to a Roth account, but 401(a)/401(k) and other employer sponsored plans do not count against the Roth IRA limit.
The IRS publishes a lot of really good information on this topic, which you can find here. It's pretty long though, so it's best used as a reference.