I am curious if I could settle at the Tenpenny Tower without blowing up the Megaton bomb and becoming evil (and if this is possible even after I disarm the bomb and fail the Mr Burk's quest to detonate it). Can I? If yes - how?
Mr Burk will only give you the key to your room as a quest reward for setting off the bomb. Even killing him doesn't get you the key.
If you are playing on PC then there are mods for this. I use one called Tenpenny Suite After Saving Megaton where you get the suite after dealing with the ghoul Roy and his gang.
Though i have not tested this, You may be able to do this by cheating using the command console. Y U NO WORK pointed out even hacking the key does not give you ownership, but with many Bethesda games, you can skip events without actually doing them to obtain the key the "proper" way by using the command console, (cheating). Through the command console, you can give yourself any quest and set it to almost any point (called the stage). there are 3 parts to this command, and all 3 can be obtained from the wiki, with 2 of them from the quest page. you would just set your quest status for the quest to blow up megaton to be right after you actually blow it up. This command can be used as a workaround for bugged quests or to skip hard/annoying parts of the quest, though use at your own risk as its impossible to predict everything that can happen.
it would look something like this:
setstage 00014e9e 60
setstage is the command to adjust your quest status, the next number is the ID of the quest Power of the Atom, and stage 60 is the stage where you collect your reward for blowing up megaton in the quest as taken from the Wiki page. (stage 50 is the stage where you press the button, so you probably have to go to 60)
The quest would likely continue as if you had blown up megaton (and people would likely also react like that), but the town would remain. But as noted, this is untested and just an idea. it may blow up megaton for you, but at least you (probably) wont get the bad karma for doing it. It may also cause other events to bug out as the state of the save file if megaton survives would be normally impossible.