Does the Vatican have any kind of impeachment procedure, or are Roman Catholics stuck with a Pope for life no matter what he does?
Boniface IX was deposed by a council. So deposition is possible. However, today a council can (apparently) only be called by a pope.
Any canon lawyers or historians with an interest in the Papacy on the Dope who might be able to give further insight?
According to the New Advent Catholic Encyclopedia:
Bolding mine. It appears that the Pope has pretty much supreme authority to decide what the Church will or will not do, including holding councils to kick him out. Of course, organizational bodies being what they are, I wouldn’t put it past the lower ranking officials to find a way of dealing with a mass murdering, slave trading, drug lord Pope or something.
Can he change doctrine arbitrarily? What if he were to wake up tomorrow and decide that all Church business was to be conducted in Chinese?
Well, there’s doctrine, and then there’s sense and process. The Pope has a great deal of power over the former, but not the latter. And even then, there’s a whole staff of people and the fact that any plausible papal candidate is basically subject to a more brutal process of scrutiny than any politician you’ll ever meet. Everybody knows what he thinks, because there’s a very small circle of elite theologians and bishops who will practically be up for popehood.
Edit: One Pope did abdicate after five months. Some consider his “replacement” to be invalid as a result. This may have been called out in Dante’s Inferno, though some think Dante was talking about Pontius Plate instead.
They managed to get rid of John Paul I quickly enough – only 33 days!
I had a theology teacher once who said that John Paul I was about to make an incorrect pronouncement ex cathedra, and so God killed him to prevent that from happening.
Boniface IX got sick and died. Are you sure you don’t mean antipope Benedict XIII, who was deposed by the Council of Constance? (There were three popes at the time, the council made them all resign and picked another one.)
There were also the depositions of Benedict IX, Sylvester III and Gregory VI at the Council of Sutri.
What about assassinations? Wasn’t there a run of Borgia-related popes?
There were two Borgia popes…Callixtus III and Alexander VI. I don’t believe that either were assassinated, though.
Actually, I did mean Bemedict IX; I got my B******* popes confused.
If a pope and the antipope of the same name should encouter each other, would thety mutually annihilate each other in a burst of energy? 
If you think they’re bad, try figuring out the Johns.
Although this example does seem unlikely, it does spark another question in my mind. What if the pope goes insane? Or like John Paul II has a condition which causes his mind to deteriorate to the point where he cannot function correctly anymore?
Can I leave that to the overage sex workers in the next sequential thread?
Even if a Pope were to establish a 25th Amendment-style (Twenty-fifth Amendment to the United States Constitution - Wikipedia) procedure for removing an incapacitated or off-his-rocker pope, his successor could immediately and unilaterally repeal it, as Josphiel noted.
Add:
[ul]
[li]Peter[/li][li]Anicetus[/li][li]Pius I[/li][li]Soter[/li][li]Eleutherius[/li][li]Calixtus[/li][li]Cornelius[/li][li]SAtephen[/li][li]Sixtus II[/li][li]Caius[/li][/ul]
<1> Office Logo | Options | Proofing | Autocorrect, enable “Replace text as you type” - Replace: Pope With: ALT+0160
<2> Insert | ClipArt | Organize Clips… | Search for “pope” - right-click on"Benedict XVI", choose “Delete from Clip Organizer”
For the win!!!