Please excuse my ignorance, but here is a 'newbie' question: Do the actors in Broadway musicals speak and sign live or are the voices pre-recorded?

It is Live.
That is the difference between a play and television and film. That is why now and then you will hear about someone flubbing their lines, tripping on stage, those little human foul-ups that can make LIVE theatre so much more fun than film. That is why some actors draw such huge crowds, to see them do it in person.

That's the answer I was hoping to get, yet after seeing 'Mamma Mia!' during the Labor Day weekend I started to doubt the notion that performances are really live. How in the world would the main characters perform a vigorous dance and sing a song at the same time. They dance, sing, then take off half of their clothes and continue performing. Where do they hide the microphones?

The microphones are usually hidden in their hair. Real Broadway purists still bemoan the use of microphones. In the old days, pre 1970's no mic's were used, so the actors had to be able to project their voices right to the back of the house. Now that's a feat.

Sometimes you can see the little thingee by their ear and the tape on the side of their head or back of their neck for the microphone
That is why they are so good! They can do all those things at the same time! And that is why it is so cool to see it on Broadway

And that's why it can cost almost $100 per ticket!

Beleive it or not, there are certain portions of some shows that are pre-recorded. There was one actress in "Phantom of the Opera" who shall remain nameless that could not sustain the high note at the end of one of the songs, and so it was pre-recorded. So there are times when certain things are pre-recorded.