by car or by train

Renting in one country and dropping in another is always expensive because they cannot rent a car in the return country that is licensed in another country. Therefore the cost of transporting the car back to its home country has to be paid by you.

Where are you staying on Lake Como? Its a big lake. A car would be a pain in the Cinque Terre. Italian train tickets are pretty cheap.

You are the 10,000th person I have seen that plans an itinerary that goes from Lake Como to the Berner Oberland. It is a clumsy train journey, but doable.

Going from Appenzel to Salzburg with a stopover at Neuschwanstein is tricky. You might want to rent a car in Innsbruck and drive to Neuschwanstein, then to Salzburg. Getting to Halstatt is also easier by car, so maybe hang on to the car and turn it in in Vienna.

For Switzerland, I suggest getting Half Fare Cards for everyone at 120 CHF pp and an 8-day Berner Oberland Pass for 250 CHF pp. I see you are staying in Lauterbrunnen AND Muerren. Just stay in Muerren the whole time, since you can see Muerren from Lauterbrunnen, so no reason to pack up and leave twice in the same area. Muerren is much nicer than staying in a dark valley.

If I'm correct, you have 7 travel days. Three Eurail passes with 7 travel days in a month would cost €792 in total, and single tickets might be even cheaper.

for folks who didn't notice this question was asked by tcplett in a slightly different way yesterday, and I and others answered your question to some degree in that question. https://community.ricksteves.com/travel-forum/switzerland/travel-from-lauterbrunnen-to-appenzell-to-salzburg

This is all so helpful - thank you!

We wanted to spend a half day and night in Appenzell to hike to Ebenalp.
Any opinions out there on whether or not it is worth it to do that stop?

We are staying in Varenna, Italy.
I looked up the route to Lauterbrunnen and wow! - that is complicated.
Any advice on the best way to do this route?

Getting to Lauterbrunnen from any point south involves at least 2 connections anyway: the Spiez-Interlaken Ost and Interlaken O- Lauterbrunnen legs are unavoidable.
To get to Spiez from Varenna, if you time it right you can do it with just one connection in Milan. So, three connections total.
If that still looks too complicated: Lake Maggiore is great too (instead of Lake Como). Base yourself in Stresa, spend 2 days exploring the lake, and you have direct trains from there to Spiez so everything is easier.