Hotel or BnB in Trondheim

I don't think there is one. Hotels that offer transportation to/from airports are extremely rare in northern Europe.

One of those exceptions:

My son stayed in an ice hotel in Kirkenes, in the way north of Norway, and transport to and from the hotel was included--by dogsled!

I loved Trondheim in August 2018.
I stayed at Thon Hotel Trondheim in the city center, conveniently near a bus stop with several lines for getting around. A small single room was only 1908 NOK ($215) total for the 3 nights.

There’s an open-air museum outside of town on bus line plus interesting museums in Trondheim itself and don’t miss Pirbadet waterfront swim/sauna facility.

This website was helpful for things to do and how to get around:
https://www.trondheim.no/en/
Includes options for how to get into town from the airport.

Enjoy!

Also, I see this is posted under Scandinavia and Baltics Reviews. You might get more replies if you move your question to the Norway forum instead.