Is it OK to send the same research proposal to several professors?
If you wrote the proposal entirely yourself, I see no problems with trying multiple universities. Do note the deadlines: you don't want to be in the situation that you need to choose between accepting an offer of a "lower-tier" university while still waiting for your first choice university to respond.
If you incorporate the suggestions of the faculty member of one university in your proposal to other universities, I'd at least let that faculty member know. Something like: "your university is my first choice, but as you know it is very competitive. Do you mind if I use this proposal to apply to other universities as a fallback"? This achieves 2 things: first of all it prevents a possible faculty member that might feel taken advantage of by using his/her comments to get into a different university. Second of all the advice you get might predominantly apply to his/her university, and might not generalize to other universities.