Renting a car vs. train

Yes, this is much too much for 10 days.

A car is useless in cities, you cannot use it for trips within the city, you have to park it somewhere (and pay). And in Venice there are no roads. A car can be useful in rural areas.
I suggest you look up each of your journeys on https://www.viamichelin.com/
This site will give routes, distances, times and costs (fuel + tolls). The times given assme no stops, so you need to add at least 25%.

This is a 10-day trip, that is 9 nights. You have 6 travel days (not including Zürich and Geneva). assume you will loose most of those days to travel. That leaves no time for sightseeing.
A good rule is at least 2 nights in each location. 10 days is maximum 3 or 4 places. Say Munich, Salzburg, Luzern and back to Munich. Depending on whether you want to visit out-of-the way places, and how many of you there are, this can be done by train or car.

Plan B, if you can gte an "open jaw" flight, would be Munich, Salzburg, Venice, Milan and fly back direct from Milan.
Or Munich, Luzern, Milan, Venice and fly back from Venice.
Or either in the opposite order.

There is a direct train from Venezia Santa Lucia to Monaco Hbf that takes about four hours and arrives before 9pm; several other trains have one or two changes. Cost is about €40/person.

Check the Trenitalia.com website (Monaco is Munich in Italian).

Yes, you need at least 2 nights per destination, so keep to the northern side of the Alps this time and leave Italy for another day. Definitely train all the way. Lausanne (midway between Geneva and Montreux) may be a better choice with easier train connections.

Munich (2)
Via Zurich to Lausanne (2 or 3)
Via Bern to Luzern (2 or 3)
Innsbruck (1)
Salzburg (2)

I have to agree - too much for 10 days. You need to drop one country - more if you are already locked into flying in & out of Munich.

Buying open-jaw flights will help loosen your schedule.

Not sure where the 4 hr train from Venice to Munich is - the fastest I found is 6.5 hrs - most are 7 hrs. We have done Venice to Munich a couple of times by train - it is a nice trip. The other option would be to fly from Venice to Munich - its an hr flight (plus the time to get to the airport and the 2ish hrs pre flight arrival time) and about $90 a person. I would be syre whether train or fly or drive to arrive back in Munich the day before your scheduled flight home.

The car will quickly become an albatross in most of the places you are going. The trains are wonderful and take you from city center to city center - ready to explore.

Unless there's some brand new unadvertised bullet train from Munich under the alps to Venice - the fastest train is 6.5 hours :-) I'm an advocate for trains all the way. HOWEVER, for travel between Munich and Venice we have found that Air Dolomiti's non-stop 55 minute flight is the to go.

The best way to see The Alps is by rental car. I'd suggest seeing Munich for 3 days and then pickup a rental car.
I'd go east to Salzburg and then south to the Innsbruck area and The Alps. Then go south thru the Brenner Pass and down to Venice.
Tirol (Western Austria) has all the mountain vistas you can handle. I'd save Switzerland for another trip.
You really need to see great European cities slowly when you're there--vs. taking on so many cities and not seeing them well.

My theory on cars vs. trains:

Trains work best for city center to city center. They skip traffic and drop you right in the part of town you want to be in. You don't have to find and pay for parking either, which can add up in cities. Trains can be relaxing and allow you to read a book, write notes or study up on the next destination. Trains don't always work well for interim stops along the way or getting to sights that are off the main tracks.

Cars work best for seeing out of the way sights and stopping and starting on your own schedule. Intermediate stops don't result in lots of lost time waiting for connections. Besides traffic and parking hassles, the down side with cars is you have to drive/navigate a car. At least one person in the car is working while others can rest, read or write.

Planes are a good choice for long legs too. In Europe there are lots of competing carriers, so flying is often the least expensive option.

I don't like travel legs of more than four hours in a day. By the time you add other things (checking in/out, packing, unpacking, finding meals, connections, getting to and from stations, etc.), four hours of travel becomes a very full day, especially if you plan an intermediate stop on the way.

My itinerary drives my choice.

Trains don't always work well for interim stops along the way or
getting to sights that are off the main tracks.

That kind of depends on the country. Germany has the most extensive track system in Europe, only 1% less dense than the much smaller Swiss system, which is the densest in Europe. France, although larger in land area, has fewer km of tracks than Germany. The systems in Spain and Ireland are pretty sparse.

In about 5 months of travel in Germany in the last 15 years, I've been to a lot of places that are off the main tracks. I generally eschew cities and large towns. For some places not served by trains, Germany has buses. I've never found a place I wanted to see that I could not get to by public transportation.

True, interim stops might take a little longer, but not that much, and you make up for that with a significantly lower cost. Since I learned how to use the German public transportation, I've saved hundreds of dollar on every trip vs having a rental car.

Yes, the quickest rail connection between Venice and Munich is the direct EC that leaves Venice at 13:50 St. Lucia, goes over Brenner Pass, and takes 6½ hours to Munich Hbf.

for travel between Munich and Venice we have found that Air Dolomiti's
non-stop 55 minute flight is the [way] to go

55 minute flight !?

I'm always amazed at how the "fly-people" blithely disregard the time and cost to get between airports and town (or maybe they just spend their entire vacation time in airport waiting rooms.

In the case of the flight from Venice to Munich, the bus schedules show it will take about 20-25 minutes to get from Piazzale Roma to Marco Polo airport. I wouldn't allow less than 1½ hour for check-in, security, and boarding. Say 2 hours from leaving Venice to flight time. Air Dolomiti shows the flight time as 1 hr 5 min. In Munich, the train into town (Hbf) takes 41-46 min, so by the time you get from the gate to the S-Bahn station, you will have spent at least an hour. So, that "55 minute flight" will really take you more like 4 hours.

For a flight a month from now, Air Dolomiti show a fare of 130€ with a checked bag rather than a 7 kg carry-on allowance. But that flight is at 6:40 AM. For a later flight, leaves Venice airport at 13:40, the fare is 176€. And it cost almost 19€ for transportation to and from the airports. For the same date, the fare for the direct EC was 49€; fares 2 month out are only 39€.

Even if the cost were the same, I would probably take the train to avoid the hassle of flying.

If you use local trains, check the schedule before you leave the station and know when you are taking the next train. By doing that you avoid getting back to the station and either having just missed a train or find you could have come an hour later and spent more time at the sight you came to see. If you know options for the next trains, you can plan accordingly. This works okay for a place like Germany where a local train comes by once an hour.

Lee: "fly-people" blithely - really??? I train all over Europe and highly recommend the trains everywhere. But yes, I do like the Air Dolomiti flights. AD is a great airline, nice attendants, nice included snacks, we love the high speed water taxis from Venice across the lagoon to the Venice Airport. Yep - I do enjoy that particular flight. But your name calling and condescension are uncalled for.