Is it weird to ask a professor you don't know if your potential project is already being worked on?
Would it be inappropriate to email a professor who's more well-versed in the field and ask if they're aware of anyone who has begun working on a similar project?

Absolutely not, I'd say. I've emailed several professors from different schools, textbook authors, and other people. While you may not get a reply (that's ok!) it takes zero effort to write and the pay off can be large. You're not going to offend someone by sending a nicely worded email inquiring about their field. If they don't care to answer they won't. So why not?