Are you allowed to alter someone else's work?
I assume so since it's so commonly done here. And it's done for much lesser reasons than fixing broken maps. Since it's done for the lesser reasons, it is not an argument against fixing maps. Anyway, this is something that can and should be addressed in the terms one accepts when uploading a map.
The map maker might not share your view on how the map should be altered, in which case you did something wrong.
We're not talking about changing or balancing maps here. Only about fixing obvious exploits, which are almost always an oversight and never a design choice. In the hypothetical case where fixing an obvious exploit violates the map makers vision, the map maker should have the right to have their map offlined instead.
What exactly should be fixed?
Obvious and easy to fix exploits. In other words, catas over map limits and players walking on map limits.
Who should do it?
Preferably the map author. But also you, me, map admins. Anyone that can be bothered.
Exploits in a map could be considered a part of the map design. Is it the job of the map admin to fix it?
You've played this game long enough to know actual oversight exploits from intentional map design. If a map author intentionally leaves something in the map that people get kicked and slain over in the servers, the map doesn't belong online in the first place.
Other map design problems, such as too much redstone, choke points, bad textures, ugly terrain, occluders, etc
Have nothing to do with exploits and don't belong in this thread. Though I would add bad occluding as something that can be fixed. But map admins are pretty good at catching bad occluding in the final check so this is not a real problem.
I would say that these issues should be first and foremost be adressed to the original map maker
This we can agree on. But in many cases the original map maker is not around anymore. If it's ok to make make altered versions of the map (and the servers are full of these) I don't see why it wouldn't be ok to fix the map and leave the credit to the original map maker.
I get your point that a map author's rights and vision should be respected. But so should the people who actually still play the game. Often abused exploits on popular maps are a major detriment to the dwindling player bases enjoyment of the game. Fixing them requires minimal effort in the map editor, and it requires no further violation of map author's rights than what is already done by allowing retextured and slightly altered versions of existing maps. Currently the approved way of fixing a broken map is adding xr_ to the name and retexturing it, and taking credit away from the original author. I'd say simply fixing the original and leaving the credit where it's due is certainly not worse.