The Jungfrau and the Valais

1) Definitely do this! Better yet, drive up to the St. Bernard pass and have a walk with them at the monastery where they are bred up in the alps...it's only about 45 from Martigny by car and there is a really great museum there with the dogs in the kennels at the back. You can walk the dogs up the path along the mountains overlooking the lake where Roman soldiers threw in offerings to the gods for safety, where Napoleon marched his troops, and where (allegedly) Hannibal crossed the alps with his elephants! The St. Bernard center in Martigny isn't that spectacular. The one at the pass is amazing!
2) Definitely get a rental car. This part of Switzerland (Geneva to Martigny) is not hilly or complicated to drive. Straight freeway all the way! Only if you go up to the St. Bernard pass will it be a windy road but it isn't that bad at all. We have found most of Switzerland to be very easy driving. Going up and over passes is a bit breathtaking but we LOVE it!
3) No experience with this...will let someone else answer.
4) You can easily take the train from Geneva airport to Martigny. There is a small train that goes from the main station up along the river. The Martigny-Bourg stop is just up the street from the St. Bernard museum. I haven't done this myself as we always have a car there. You could also take a taxi from the station...it's only a few blocks. There is a nice little Roman amphitheater next to the museum as well - added bonus!
5)It's a pretty long drive from Bern to Martigny...you could definitely do it but maybe consider spending the night somewhere to make it less stressful?

Honestly, between the car and the train, I would opt for the car. In my opinion, trains in Switzerland are for seeing the views on the more spectacular train routes. For where you are going though, and for what you want to see, I would get a car and give yourself the freedom to stop and see all the great places you'll come across along the way. It will also save you time. Gas is very pricey though so be prepared to pay!

"3) If we rent a car it means we'd have to pay to park it in Lauterbrunnen because we hope to stay in Wengen."

Your hotel may give you a parking voucher, but it depends on where you stay.

It is very easy to take the train to Martigny. I walked from the train stn. to the St. Bernard Museum, the dogs are so friendly, then to the Roman Arena. A block away is the Fondation Pierre Giannada, a wonderful art museum, with statues/art in the beautiful gardens as well as in the building. I was coming from Zermatt and heading to Lausanne.

You can easily do all of this without a rental car. Lots of public transportation to the areas you have identified. Land in Zurich and 3hrs later, you are in Wengen. You do change trains 3 times at Bern, Interlaken Ost and at Lauterbrunnen. These are really easy train changes. To get to Martigny, it's a 90-minute train ride from Interlaken Ost with train changes at Spiez and Visp.