I live in Colorado and wish to install a basement shower and use the existing floor drain in the basement to connect the shower basin drain to it. Is that OK to do? What problems might I encounter? How do I know if one exists? If none exists, what are the code requirements? The house was built in the 60's?

See this post by our resident Plumbing Expert:
https://www.askmehelpdesk.com/plumbing/hooking-into-main-line-24911.html#post113517

You'll see that the answer is yes and no. Yes, you can connect to it, but you still need to vent the shower drain.