I’ve seen the closing time for the Sistine Chapel listed in different places as being at 4:00, 4:30 and 6:00, and I don’t know which one of these times is correct. If you know the closing time, please clue me in. Thanks.
Hopefully this website will answer your question. Looks like maybe it's open until 6:00 most days, but the ticket office closes at 4:00. That may be what was confusing you. Scroll down the hours are near the bottom of the page.
If you don't have a ticket ahead of time, your chances of seeing the Sistine Chapel will be slim or you will wait for a LONG time. Also, watch out for pick pockets here. Sad to say, but true.
This is the official website for the Vatican Museums, and the chapel is IN the museums. The page I'm giving you is for the hours:
http://www.museivaticani.va/content/museivaticani/en/info/tutti-gli-orari.html
The noted time you need to pay attention to is final entry to the museums at 4:00 PM on normal visiting days. The chapel is a brisk 30-minute walk from the entrance. and they start clearing the museums 30 minutes before closing.
And yes, I'll strongly second the advice to buy tickets in advance.