Any question answered...

Stay away from Lutefisk! Definitely have some brunost, they have different strengths but I'd recommend trying the darkest one first. Also whale is still in season and delicious, cod tongues are also quite good if you're into weird fishy shit.

Air bnb is prob best value for finding somewhere to sleep, east Oslo is cheaper in general. Grünerløkka is young/trendy/hipsterish, Grønland is the immigrant area, Gamlebyen is one of the last city centre neighbourhoods not to be gentrified. West Oslo is generally posher but closer to Bygdøy which is nice.

As for things to do it really depends on when you're coming, the weather can make a big difference. Don't miss Vigelandsparken, Frognerseteren, Bygdøy, views from the Opera house roof and Akershus fortress walls. Also some of the museums are free on Sundays.

Full disclaimer I sometimes work as a guide at vikingbikingoslo.com but the highlights tour is a great way to get your bearings if you're arriving before the snow does.