Did the Boston Patriots play at Fenway Park?

Answer 1

Yes, and other places. They never had their own stadium until they moved to Foxboro and became the New England Patriots.

The Wikipedia article says this:

The Patriots time in the AFL saw them struggle most years as they never had a regular home stadium. Nickerson Field, Harvard Stadium, Fenway Park, and Alumni Stadium all served as home fields during their time in the American Football League. Nickerson Field belongs to Boston University. Alumni Stadium is at Boston College.

Answer 2

No. RFK Stadium is in Washington DC, and was the home of the Washington Redskins before it was replaced by Fedex Field.

Answer 3

They played six home games at Fenway Park and played the Jets in a "home" game at Legion Field in Birmingham, AL because Fenway Park was not available at the time.