Of course. Twitter allows sponsored tweets and advertising links in tweets. If your followers like them or not is another question.
Maybe if you are popular with thousands of followers like John Chow you may get amount of income/profits.
I thought Twitter recently automatically shortens links in a t.co format. Although they are display in full, the link target is shortened with this service.
Why not but you need to be very targeted and have good number of genuine followers. If your followers are made up of Internet marketers, then guess what!!! Most of them will use their links to buy That happens a lot of times