What are some of the similarities between Doctor Faustus, Richard III, and Macbeth?
What a great question! These are three very deep and rich texts you are comparing, but I would say the most clear comparison we can draw between them lies in the way that each of these three characters pledge themselves to evil and how the rest of the play charts their inexorable descent into perdition. Perhaps this is most clearly shown in the character of Doctor Faustus, who makes a literal pact with the devil ensuring the loss of his soul after a period of time. The rest of the play therefore shows the panic of Faustus as he realises that, however much power he has and whatever he is able to do, that power is not his own and only brings his eternal damnation that much closer.
If we consider the characters of Richard and Macbeth, we can see something similar. Both are characters who deliberately engage in evil acts in order to gain temporal power in exchange for sacrificing eternal salvation. This decision is made early on in the plays, and the rest of the action shows how they continue to mire themselves in sin after sin. Note in particular this famous quote from Macbeth from Act III scene 4 that describes his state:
I am in blood
Stepp'd in so far, that, should I wade no more,
Returning were as tedious as go o'er.
Macbeth identifies that he is so guilty now that it would be just as hard to try and go back as it would to carry on, so he might as well carry on.
These three plays therefore present us with characters who effectively damn themselves at the beginning of the play and then consolidate that act as their inevitable end approaches. They each exchange eternal salvation for temporal power, and are shown to be worse off for this exchange.