I think we do have the same tank!! My Toby loved it (before he was transferred to the 10 gallon lol).
As far as his behavior goes....he may just be shy, especially since you've only had him for a week. Some bettas are much more feisty and eager to explore. I would keep an eye on him for now, he may just be adjusting (especially with the filter mishap lol). As long as he seems normal (eating well, swimming around) then you probably don't have anything to worry about. As far as his fins...he may have been trying to bite them to relieve some of the weight. Especially from the filter flow. My VT Leo did that for the longest time and I could never figure out what it was. It was a combo of stress from his huge 3 gallon tank (he's not a "big tank betta" lol) and feeling stressed from the filter (also a Whisper). As soon as I moved him to a 1 gallon QT he has stopped biting his tail....very odd confusing situation but without the filter he is definitely much better!
And such a handsome betta by the way (;