To answer your first question, apply to many as you like, apply to the ones you would like to go to. if you have stellar grades and scores, than you can be more choosy about where and how many you want to apply, and well if you're grades and scores aren't so top notch (but still good obv.) then you can't be so picky.
Secondly, colleges usually do find out about where you apply, but only because you tell them on your secondary application. Of course, you can lie about it-but that's an ethical dilemma you must consider. Furthermore, there is no such thing as being "desperate" when applyin to med school. We're all desperate!!! After all you're trying to get into a field where many people attempt but very few follow through. Good luck in your future endeavours