The short answer is no. There's no single list or map like you describe. There's also no simple answer according to the hut type. Staffed huts will always be unlocked when they are staffed, but at other times of the year may operate as a self-service (or occasionally unserviced) hut, which is likely to be locked but some may be unlocked. Self-service and Unserviced huts fall into locked and unlocked. Add to that some locked huts turn out to be unlocked when you get to them (but if it's badged as locked I wouldn't want to risk getting there without a key).
You do need to look up the details and dates for individual huts. But, in case you hadn't realised, the UT map you linked to is clickable, so you can zoom in to an area, click the hut symbol to get a pop-up, and then click the hut name to open it's info page (if you open in a new tab you can easily open all the huts in an area).
And (not sure if you've sussed the Norwegian words)
stengt - closed
betjent - staffed/serviced
selvbetjent - self-serviced
ubetjent - noservice
ulåst - open (no key needed)
DNT-nøkkel - DNT key
The DNT print an overview map very similar to the UT one, but really the UT map is the quickest way I've found of researching the information.