Do I need to go to the doctor if I was shaving my "down stairs" with a razor that had a little spot of rust?

No. Touching rust if not a problem. People use, handle, and play with many metal objects or tools that have rust on them. The "stepping on a rusty nail" concern occurs because germs are pushed through the skin wound, deep enough to cause an infection. However, razors shave better if they are smooth and rust-free, so a rusty razor should be safely disposed of.