Changing planes in Paris

Only 1 hour between flights.

That sounds too tight to me - especially at CDG, an airport not known for its efficiency. This is all on one ticket, I presume, so they should get you there if you miss your connection (quite possible - my recent flight on Delta from Seattle to Amsterdam left 2 hours late...). Are there other/later flights to Montpellier that same day? You will want to investigate what options would be if you do miss that connection.

Do we go through customs in Paris?

Yes, that's what I would expect.

Will our bags be checked through to destination?

Yes, assuming the connecting flight to Montpellier is on Delta (or one of their partners that Delta "interlines" bags with). But you will need to claim your bags at CDG, go through customs (and passport control), then re-check them for the onward connection.

With just one hour for all that, at CDG, I would not bet on making your connection. It's possible you might, but consider exactly how much it would mess with your plans if you missed it. If it were me, I'd want a minimum of 2 hours, more like 3 to ensure a connection there. Depends on how lucky you're feeling...

Assuming that Paris is your first European/Schengen stop (sounds like it from your question), you will go through immigration/passport control in Paris and Customs in Montpellier. Immigration is the time-waster, not customs. 1 hour seems unlikely absent some help from airline personnel. If you're travelling on one ticket, it is the airline's responsibility to get you to your destination on the next available flight if you miss your connection. And your bags will be checked through only if you're travelling on one ticket.

I see that Delta is selling that one-hour connection, code-shared with Air France. I have also, on separate trips, had my SEA-CDG flight delayed by an hour, and another time missed a connection due to passport control taking 45 minutes. Immigration/passport control is a wild card. I believe they're now more alert to helping passengers whose ticket shows a short connection but you'll have to be proactive. If you miss the one-hour connection, they should put you on their next flight 3.5 hours later, if there's room.

Even if you make it to the internal flight in time, your checked bags may not.

I flew JFK to Paris to Montpellier in 2010. I had allotted 3 hours for the transfer, but my plane was about 2 hours late taking off, so I was sweating the entire flight. I knew that since I was on one ticket, they'd have to get me to Montpellier eventually, but having read all the horror stories, I was sure I wouldn't make my connection.

I got to Paris and, after immigration, ran to make my connection (2E and 2F are across from each other). When I got to the security checkpoint, the screener noted my distress. She urged me not to worry, saying I had plenty of time to relax, get a coffee, and meet a girl. Yes, she actually said that. I guess in Paris, there's always time for romance. I would have preferred to meet a boy, but alas there was no romance for me. However, she was right, in that I did indeed have time to get breakfast before my plane took off. My checked bag made it too.

So, you may well make it. If you don't, there are several flights flights per day from CDG to Montpellier.

Assuming you are all on one ticket, you will go through immigration in Paris, and customs in Montpellier. Immigration can be quick or slow, and how long it takes cannot be predicted. You then go to your next gate, changing terminals and going through security. If you're on one ticket, your bags will be checked through to Montpellier; you will surrender them in SLC and not see them in Paris.

In Montpellier, your checked bags may arrive at a different place from everyone else's, since most people on that flight will not be going through customs and you will. If you can't find your bags, ask. After getting your bags you go through customs, but this is just a matter of walking through the Green Channel Nothing To Declare line. Unless you're "randomly selected for additional screening" (possible but rare), you're done.