Should I send professors money for illegally downloading their books?
With that in mind, would it be appropriate to simply send those odd 10 dollars to the author of the book that I am illegally downloading? 


First, I would be concerned about your legal exposure. You would be effectively admitting piracy.

Second, you cannot unilaterally change the terms of sale. When the professor published the book, they agreed to sell it through the publisher in exchange for certain terms. The legal and (in my view) ethical options are to accept or decline these terms; you cannot invent and execute your own terms instead, even if they seem reasonable. In short, this is a rationalization. (That said, I personally am sympathetic to your concerns about publishing companies exploiting college students.) 

Thus, the "appropriate" thing to do is to buy the books through legal channels.


  If you are a professor who authored a book, how would you feel about this?


Not a professor, but I have taught and written a book with thousands of copies sold. I'm sure I am losing money due to piracy, but I have never received a payment like you describe. 


Realistically, if I were to receive cash anonymously, I would probably chuckle and pocket the cash, or maybe set it aside for a few years to see if anything came of it. 
If I received money from a known student, I would be very concerned about the appearance of impropriety, and would not accept it. I would also be concerned about whether I should report the piracy, though I probably wouldn't.
If this "caught on" and I was receiving a non-negligible amount of money from many pirates, I would have to talk to the publisher and seek guidance.