Norway on $2000 / day ?

No question that Norway is expensive. $350 USD for 5 persons would only be possible if you camp as in a tent. Norway has a concept of Allemansrätten - 'Everyman’s right' or freedom to roam. Camp where you want as long as you respect others. http://www.nordstjernan.com/news/nordic/2540/

Allemansrätten, the Right of Public Access, freedom to roam, or
"everyman’s right," is a term describing the general public’s right to
access certain public or privately owned land for recreation and
exercise.

If you need/want beds and meals, try a DNT guided trip. As an example: https://english.dnt.no/aktiviteter/826493/
With transportation to/from Oslo and 8 days/7 nights full board, NOK 7.837.- from/to Oslo (plus Additional non-member price NOK 700) or $1040 per person = $150 per night per person.

This is without doubt one of our most traditional hikes! Visit the
heart of Jotunheimen National Park and experience one of Norway’s
highest summits Glittertind. The scenic walk across the Besseggen
ridge, is also a part of the program. This will be a fantastic
experience if you are in good physical shape and sure-footed!
Accommodation in staffed lodges.

When I visited Norway, I did it with a rental car (as part of a longer trip) and was quite happy with scenic driving opportunities. There's no need to take the Norway in a Nutshell excursion when you have your own wheels, although the driver does have to watch the road.

Youth hostels in Norway are quite nice and often have family-sized rooms. Camping cabins are another budget option; you might need your own sheets but not tents and other equipment. Ricks' Scandinavia book lists several of each in the Sognefjord area, for instance, more than he would in other parts of Europe.

Good advice from Laura re Norwegian hostels. They are basic and not exactly in city center but not your image of a "youth" hostel sharing bunk beds with young folks. Hostels have private and family rooms. During our economy travel days we have stayed at the Oslo Haraldsheim and had private room during one visit and an en suite family room (with just the 2 of us).

http://hihostels.no/en/hostel/oslo-hostel-haraldsheim/

Oslo Hostel Haraldsheim is a pleasant hostel that offers budget
accommodation at Grefsen in Oslo. Nice common areas indoors and
outdoors (Internet access and BBQ) and a great view of the city centre
and the fjord.

The hostel has 1-4 bedded rooms, many with private bath. Breakfast
buffet is included.

For a bit more per night, the "budget" Hotel Thon chain is something to consider. The breakfast buffets are a big step up from hostel breakfast buffets.
http://www.thonhotels.no/
BUDGET Sov godt til drømmepris: http://www.thonhotels.com/hotels/consepts/budget1/

Sleep well at low rates

Thon Budget offer functional hotels with everything you need for a
good night's sleep. At our Thon Budget hotels you get a comfortable
room, a healthy breakfast buffet and free wireless internet access.

Eat a big breakfast, snack on a Norwegian pølse (hot dog) for lunch, and find a kebab (or a Peppe's Pizza) stand for dinner. Stock up on alcohol at the duty free before existing the secured section of the airport or see if you can find beer/Øl at the Coop market.

https://en.wikipedia.org/wiki/Peppes_Pizza

PS When a hotel has a room for one, they mean one person, not one bed and its a one person bed.

Good replies. I kind of dread hostels based on experience but have never had a private room in one.

My usual standby hotels for free nights on points (this really can keep down travel costs), IHG (Holiday Inn, Intercontinental, Crowne Plaza) and Marriott are absent from Norway and Sweden (except 1 Marriott in Stockholm) so I will have to scramble for any free stays. Lots of Choice Hotels all around Norway so if I am willing to mattress hop on work travel (8000 pts every 2 stays, so if I am willing to split a week of work travel by literally walking across a parking lot separating a Comfort Inn and a Sleep Inn alternating 1 night in each can rack up 16,000 points/week that way). But a bit perplexed that most hotel rooms in Norway have a double bed or 2 twins? Then would need 3 rooms/night!

So maybe 1 night in Alesund and then 3 nights on a fjord?

But a bit perplexed that most hotel rooms in Norway have a double bed or 2 twins? Then would need 3 rooms/night!
Key in hotel bookings is the need to specify the number of people. As noted in my earlier remarks, a single room is a one person room with a one person bed. You might be able to find one room for three persons (2 beds) plus room for two persons (with two single beds).

Just checked the Thon Oslo Astria for a sample date in August. For 3 persons night of 25 August:

STANDARD TRIPLE/FAMILY JUST 3 LEFT Double room with twin beds (2 x
80-90 cm) and additionally extra bed or sleeping couch, suitable for
up to three (four) persons. Thon WebDeal NOK 1,192 per night* Select
See details and conditions. Thu 25. 1,192 NOK Fri 26. Special offer,
bookable on Thon Hotels website only. Rate is per room and includes
breakfast, wifi and VAT. Rate requires payment upon booking.
Cancellation within 24 hours after the booking was made. If the
check-in date is the day after the booking was made, cancellation must
be done within 4 hours after the booking was made. The special offer
is non-refundable and cannot be changed once the cancellation deadline
has passed.

The Choice website is surprisingly funky, for example I can get a room in Bergen for three with this strange message:

ROOM 1
1 Double Bed, No Smoking
(Max Occupancy: 5)
2 Adults , 1 Child
Room Charges: 6,000 Points & $75.00 USD

How can 1 bed sleep 5?

Or for another hotel in Bergen I get a button to select a room, but when I do get this message:

We're sorry, an unexpected error occurred.

Or a third hotel gives me this:

An extra bed is required for the number of occupants you have selected, but no extra beds are currently available.

For every night even though they can't be out of extra beds every night, It can't always be true.

OK, looking at family rooms in hostels, there are a very few with rooms for 4 or 5 people in the fjord area, running about $185 ouch. But reservable now on website with persistence (lots of clicking).

Lots of house and cabin rentals also available, in the $200-$400 range so no more than 2 (or 3?) hotel rooms.

Mostly confirming the trip is reasonable, maybe raise budget to $600/day, so can book flights (cheap at $450 per direction) and reserve car, then get to the details of staying where and seeing what later on.

Norwegian hotel prices often include buffet breakfasts. A Norwegain buffet breakfast in substantially more substantial than the typical American motel breakfast buffet where hot breakfast often means a do-it-yourself waffle machine waffle. And Norwegian hotel coffee is on par with American coffee kiosk drip coffee if not stronger. You can pack away enough calories to carry you through the day.

Speaking of waffles, I suggest splurging on a Norwegian vafler (waffle) with cream for an afternoon snack. They are no where near an American breakfast waffle.

Spent a lot of time looking at vrbo and airbnb and seeing that renting vacation cabins is no more expensive in Norway than other places in Europe, and neither is renting a car. Good news, it won't be so terribly expensive after all.

Here's one person's view about Norway costs - from a Swedish perspective.

I was coming into Chicago on the El from O'Hare, and struck up a conversation with a Swedish woman. She said that her family loved going to Norway, but they had to take a lot of steps to cut costs. They brought a cooler full of food, since food was so much more expensive in Norway. They filled up the tank in Sweden before crossing the border and tried to do the whole trip on one tank of gas, since it was so much more expensive in Norway. They brought camping equipment, since they couldn't afford Norwegian lodging prices. At this point I interrupted and asked, is Norway really that much more expensive than Sweden (a country not known for its low prices)? She said absolutely yes. I then asked, was all this effort really worth it? She said absolutely yes, Norway was far more beautiful than Sweden.

I can vouch that prices in Oslo are definitely higher than in Stockholm for many items. Rick's Scandinavia book has various money-saving tips for both countries that I found very useful. A big one I remember is that Monday through Friday, Stockholm restaurants have lunch specials at a very low price. You can have a "real meal" at lunch and then a sandwich or similar at dinner, rather than the reverse, and save a lot.

We spent a month in Norway last summer. Although we were there on a home exchange there is the possibility of trying Airbnb listings...this could be a great option for you as many listings will fall within your budget. This can also cut down on your food costs since you can make breakfast at "home" and even dinner if you want. Eating in restaurants in Norway is very expensive...grocery stores are not much more expensive than the ones in the US, at least where I live. Even if they seem expensive to you they will be FAR cheaper than eating out.
We spent most of our time in Norway in and around Alesund...the area of the Sunmore Alps, the Geirangerfjord, and the Trollstigen Pass are stunning. Beautiful ocean and coastline too with islands and lots of wildlife. It would be a perfect place to spend your 3-4 days.

Thanks for the additional comments. Unfortunately we may be pushing the trip forward to 2018 due to scheduling issues, but that just gives me more time to save up Choice Hotel points and plan more thoroughly.

Surprised that legacy carriers are offering prices lower than Icelandair for next June, Delta one stops into Oslo return Stockholm with free Amsterdam stopover for $805, and American only $670 for the same itinerary but with 2 stops each way so not desirable.

We are also a family of 5 and it's really expensive, esp with teen appetites and need for more space. If you're not wedded to a land tour, you might consider a cruise on Pullmantur. (Disregard UK prices on website, US prices are quite a bit lower.). I was considering them for next year but we're probably going to UK/Italy instead. They are a Spanish cruiseline and 3 star - definitely not in the same league as the major cruise lines. However, they have some very affordable cruise prices for Norway and the Baltics. I priced out their 7 night Norwegian cruise with 2 inside cabins and it was under $4,000 (just over $500 per night). For a more spacious balcony or suite with an inside, it was about $5,000. Definitely cheaper than the Hurtigruten and much cheaper than other standard lines. It's hard to find information on Pullmantur but cruisecritic has some and here's a good recent review from an English speaker.

I agree with the comment about using Airbnb. We are going to Norway next September and have been checking for Airbnb apartments in both Oslo and Bergen. There are many options available and they would work great for a family of 5. We traveled for 13 weeks in Europe last summer with our 2 grandchildren and used Airbnb almost exclusively. It gave a more space and we always ate at least breakfast in our apartments.