Vatican Re-entry?

I do not recall needing a ticket to enter St Peter's. Your ticket will be for the Vatican Museums (of which the Sistine Chapel is a part). If you leave St. Peter's and return, whether the same day or any other time, I think you would enter St. Peter's through its regular visitors' line, then go to the line for the dome.

You can't re-enter the Vatican museums but you can go into St. Peter's as often as you like. There is no ticket for the church and the entrance is separate from the Vatican museums.

Here's the web site for St. Peter's. There's a map so you can see where the museum entrance is and where the church is located. You enter the church from St. Peter's square (Piazza San Pietro).

http://saintpetersbasilica.org/touristinfo.htm

>>>Here's the web site for St. Peter's. <<<

That's not the website for St. Peter's and it's not the official Vatican website. It's just a site by someone that has a big interest in the Vatican. Here's the official website (note the .va at the end for Vatican).
http://www.vatican.va/phome_en.htm

>>>You can't re-enter the Vatican museums<<<

Yes, you can re-enter the Vatican Museums the same day with your ticket (or at least you can on individual tickets and the Vatican tours that the Vatican offers).

Climbing the dome requires a separate ticket and is not included in any other tours or tickets. You might find a longish line for that. Tickets are about 6€.