Hi Jane -
Welcome to My Swiss Alps.
There are a lot of questions in your post. ;-)
I'll start on answers to some of them.
Luzern is one possible good base, due to the variety of places one can reach from Luzern.
Another good base is the Jungfrau region....possibly Interlaken, although it is not really close to the nicest places, which cannot be reached by car...prohibited. I'm thinkiungof the viallages of Wengen and Mürren. You can take a car to Grindelwald or Lauterbrunnen.... and park it.
You have noted that you plan to stay outside of cities. That is wise. Parking in the cities is very expensive and scarce.
The way that they keep spaces available is to raise the price until many people just don't use the garages. ;-)
You are probably aware that mush of the most scenic parts of Switzerland cannot be reached by road.
For example, in the Jungfrau region, you can park at Lauterbrunnen, Stechelberg and Grindelwald. Or, Interlaken.
I don't know about the density of rail and bus transport in Norway, but Switzerland has a dense network of frequent and effective public transport. It includes effective luggage transport, which works particularly well if you are based in a location for more than one or two days. (Fast luggage transport is available, but more expensive.)
When I travel in Switzerland these days, I rent a car, and use it to reach trail heads or get to a village where I will stay for a few days. Often, the car then remains parked for days in a row.
What I do not understand is where you would park a camper. That doesn't meanmuch. I've simply never looked for places to do that. I have noticed "No Parking " signs even at remote locations. And, even at the end of long winding mountain road to S-charl, on the backside of the Swiss National Park near Alp Tavrü the parking area has a "Zentral Parkuhr" - a fee for parking, with a machine to give you a ticket to leave on your dashboard.
I do know that there are campsites available.
This link might get you started. Check the links listed at the end.
The menus in this map are helpful to understand transport and parking.
map.search.ch/?pos=637832,160552&z=32 &poi=bergbahn,halteste lle,parking,veloland,z ug
A good map for hiking trails and altitudes is:
map.wanderland.ch/?lang=de&route=all&bgLa yer=pk&resolution=250& X=664750&Y=219500&laye rs=Wanderland
In this link I have turned on "accomodation" and you can look along alpine trails
map.wanderland.ch/?lang=en&route=all&bgLa yer=pk&resolution=250& X=664750&Y=219500&laye rs=Wanderland%2CAccomo dation
Note that the Switzerland Mobility web site, in addition to the map just linked, has a wealth of information about hiking trails. It takes a bit of exploration to find your way into it, but once you have atrail located, you are presented with a detailed map, an altitude profile, and accomodations and points of interest along the trails.
There is longstanding traditon in Switzerland of "Mountain Hotels" many of which can only be reached on foot.
I am trying to think of a central "directory" or link to guide you,and this may give you a start:
Here are random thoughts about some places with high mountain hotels. Some don't have much hiking nearby, others do.
The Rigi - Rigi Kulm, Rigi Scheidegg, Rigi Kaltbad, and other places on the mountain.
Kleine Scheidegg - a very traditional mountain hotel. often used by climbers.
Here is a good example along the route of a strenuous hike-
Here is one that is off the beaten path....and famous:
Another, picked at random: