Drive from Trondheim to Bergen, how many days

Alesund, or the surrounding area, is the most central to all of the places you list. The only stave church within reach of Alesund is the one at Kvernes but you can combine this with your drive over the Atlantic Ocean Road...fairly long day but it can be done. This little stave church is well worth a visit!
Each of the other things you list can be done as day trips from Alesund. I know of a nice airbnb in the town of Sykkylven, which is about 30-40 minutes outside of Alesund on a very pretty fjord. We had a home exchange with the people who own it - it's a self-contained apartment on the top floor of their house. PM me if you want more info! It puts you that much closer to both Trollstigen and the Geirangerfjord.

I think I would choose 2 bases for your time and distance.
No matter what I don't really see one or even two locations that don't require a fair amount of backtracking to see what you want, unless you move bases nightly.
This is a rough plan that would need some research to tweak but at least may get you started.

First day I would drive south, take the Atlantic Ocean Road and stop and stay in Alesund. This is on your list anyway and will be a full day to get there. Stay 3 nights
2nd day visit Geiranger, drive to Hellysett and take the car ferry in.
*** alternatively I think you could catch the ferry from Aselund and leave the car parked today allowing you to rest from driving***
3rd day drive East again but drive to Andalsnes to the Trollstigen and see the mountains around Romsdalen (you could squeeze this into the end of day 2 if you do take your car to Geiranger but better not to rush it I think)
4th day drive to either Balestrand or Sognalsfjorda ; that general vicinity and find a place with a nice cabin by a fjord ; I am sure you could find such a place in that overall area. Stay 2 nights
5th day: visit the nearby Urnes Stave Church ; one of the most famous in Norway and still have time for a boat ride on the fjord in the immediate area
6th day: Drive to Bergen ; stopping off a famous waterfall or two along the way

Splits up the journal off 13 hours or so into 3 ; 5 hour drives roughly so I think makes the most sense and ticks the items off your list.

Those are good ideas. Didn't mention that we are 4-5 so hotels at 2 rooms/night are budget busters.

I was thinking of another plan: 6 nights SE of Ålesund somewhere on a fjord, the cabins are very reasonably priced (not like hotels!) and cook or picnic avoiding eating out. This allows us to visit Norway on a France budget. Then drop down to Oslo and Drammen (visit with relatives) and visiting a stav church on the way. Then train to Stockholm for 3 nights and fly home.

So skip Bergen and the whole Norway in a Nutshell stuff! Can I do that? I'm hearing no, that Bergen and the fjord cruise are must sees, that these can't be missed. But I'm liking the north stuff, and maybe Trondheim and Ålesund are enough without seeing Bergen?

Yeah hotels in Norway are very expensive.
I did some research on cabins like you are talking about for the area around Andalsnes ; none of them will have fjord views in that area though ; mountain views = yes

if headed to Oslo and not Bergen that could be a good place for you as a base. Oslo is closer than Bergen to Trondheim so you really don't need 2 bases like I think you have with the other plan.
Dramatic scenery, Trollstigen right there, Rauma River runs through the area with mountains on both sides, Geiranger not far away for a day trip, Alesund not far away, even the Atlantic Road is doable as a day trip without changing bases.

There are a half dozen options of campgrounds like this in that area
This one stood out to me as the best option.
http://www.mjelvacamping.no/

Camping is tough because of all the equipment, and then there are the rain/noise issues.

Thinking that if I keep alcohol, restaurants, and hotels down to a minimum Norway won't be such a killer on the wallet. Was able to reserve a car for a week from Avis Trondheim to Oslo for $500 which is still more than in would be in France but less than Mexico.

Consider stopping at Dovrefjell national park. You will drive through it on the way to Dombass. Stop at Oppdal. It isn't much but is the base for recreation and slate tile industry. Take a day trip to see the local musk ox. The hike does require some fitness. Straight uphill likely in rain and mud. Local guide services are near the train station. Spectacular views.