Should I ask my research advisor for extra hourly work as a gap-year research assistant (not a student)? I'm currently working as a post-bac researcher with my undergraduate advisor in the US, but lost my funding last summer. So, I have been doing the research as voluntary work and work odd jobs that are not contributing to my career just to pay the bills. I'm in a tight financial situation where earning some extra money would be good.

Since I am not a graduate student, would it be reasonable to ask my advisor if I can be paid for some side research and TA-related work (e.g. grading papers for a class)? Does this seem like atypical request to a professor?
There's nothing wrong with asking.  However, I suspect it's unlikely you'll get it.  Such work is often restricted to enrolled students only.