Looking at booking a flight to Munich - I can either do a direct flight from the US to FRA then connect to MUC or book a direct flight to FRA then train to MUC on one ticket (sort of).

I have a few questions for anyone that has done this.
1. Do I pick up my luggage as normal in FRA and head to the train (essentially like I booked two separate tickets)?
2. How much time do I need to plan on for this connection - I know things can vary dramatically - but it would be a Saturday or Sunday morning - arriving in FRA at 6 am.
3. If I take the plane connection (allowing 2 hours to connect in FRA) I get into the MUC airport at 9:10a. If I take the flight + train I get into Munich central train station at 12:10am (could get in earlier at 11:07 if I take a flight to train connection of 1.5 hrs). The question is - how much time and it takes to collect luggage in MUC, clear immigration, then take another train and/or taxi to our hotel (to be determined) in MUC - I could see that easily being another 2 hours - right? (if not more)- so all things considered, the flight + train could be very comparable in terms of total travel time? plus it is a little cheaper - $40 per person plus cost of taxi and one less segment (assuming we take the train from MUC to the train station vs a taxi).
4. It sounds like train tickets are not for a specific train, just any train on that route that day - so I definitely could go for the earlier flight
https://www.lufthansa.com/de/en/rail-and-fly
5. I believe the price quoted is for a second class seat on the train - (assuming that price is built in - hard to tell) - if I want to upgrade to first class - it is 30 euro more - how do I do that?


thanks again all