If Neal Cotts has, say, 3 options left, and Jamie Burke also has 3 options left, and we send Burke down to bring up Cotts, have Cotts start a game, and then "send him down" though he technically is not demoted(he is just taken off the roster and kept in the dugout, much as he did last night) and Burke is brought back up,would an option be taken up every time that is done?

Options are by years, not by how many times you get sent down. So if a guy has 2 options left, he can be sent up and down in 2 different years, not 2 different times in 1 year or anything...

So if we wanted to, we could send Cotts down after the game, bring Burke back up for 4 games, and then when it is Cotts turn in the rotation again, send Burke down and bring Cotts up?

I imagine if they wanted to do that, they could..but Cotts and his pitching position in the majors/minors would be totally fubar. For that reason alone, they probably won't do that

If they send him down an Option is Burned leaving him with 2. Next year if he gets send down another option burns

Airfare would be prohibitive.

you have to spend 10 days in the minors after getting demoted before you can get called up again.