Bayit (בית) is the Hebrew word for "house". The title refers to the 1983 Talking Heads song "Burning Down the House".
Quagmire looking in the mirror and seeing another person, this being Scott Bakula is a reference to the actor's show Quantum Leap in which his character Sam Beckett looks in the mirror and sees his host, another person.
Peter and Quagmire working the pill line is a parody of the I Love Lucy episode "Job Switching" in which Lucy and Ethel Mertz work a conveyor belt of chocolates. 
Stewie says that Lois tricked him into visiting prison by telling him that she is taking him to Carvels. Carvels is an ice-cream retail chain.
Peter decides that the group's alibi is going to be that they were watching The Beatles at The Ed Sullivan Theater. A cutaway shows The Beatles performing on the show which was broadcast nationwide.  During a song, Peter throws his dirty underwear at them. This references the mass hysteria among teenage girls when The Beatles performed on The Ed Sullivan Show.