Okay, I have looked into a few of these things. Allow me to share my notes.
1) "With BuildTools, it's possible to downgrade structures all the way to twig. This creates gaps that allow players to see through and look under the map."
It appears that BuildTools does have an API for overriding whether or not you can downgrade an entity when using the /down all [wood/stone, ect..] command. However, it doesnt appear (at least that I can find) to have one for the single downgrade entity. I will be implementing the fix for the "all" version, but unless there is an API for the single - I can't do anything about that at the moment. If it is an issue, perhaps you can limit the downgrading to be wood at the minimum? That way they at least cannot see through the walls. Luckily, it appears that upgrading works fine (Basements prevents it).
2) "Hatches can be picked up using BuildTools and are then added to the player’s inventory."
BuildTools has an API for this, I will be implementing it with the next patch along with a config option called "Allow Hatch Removal". It will default to false, but if true it will allow plugins such as BuildTools or RemoverTool to remove the hatch. If false, you will not be able to remove it.
3) "Player corpses do not spawn above the map but instead appear underneath it."
I couldn't reproduce this one. I tried dying in a basement, and the corpse was on the floor of the basement. I also tried dying, and then destroying the basement, in which the corpse bag floated to the surface. I will try to play around with this one more.
4) Additionally, there's no error message when a player tries to place a hatch without having the required resources in their inventory, which is causing confusion.
Yup, this is a straight up bug, there is supposed to be a message, it will be fixed in the next patch (coming very soon).
I will be DMing you a version with the current iteration of these fixes for you to verify - I appreciate the detailed bug report!