I think they are random but they happen at the bottom on your vault in the rooms that are just above the dirt or you could keep rushing rooms and hope you get a mole infestation, if you want to intentionally cause one that is.
Catch the Mole. take him to the shelter where he will be humanally put to sleep and he will no longer grow.
It is not a good idea. You are likely to cause a large scar where the mole is. And if the mole should be cancerous, it could cause it to spread.
they go in the badgers house for shelter
It digs a hole in the ground (like an ants home).
. . . the only reason i can think of for you actually wanting to find a mole rat is the quest for Moria where you have to kill 10 of them as an optional quest. In that case, you should already have the tepid sewers on your map.
Only if its diabeetus!
Not normally.
Mr.Badger
There are no enimies inside the vault itself but there are a few mole rats in Smith Casey's garage, which you have to go through to get to vault 112
A fossorial adaptation refers to a fossorial creature that has adapted to digging and burrowing for shelter. The badger and naked mole rat are fossorial.
A hole in the ground used for shelter
Take notice to see if the mole changes over time or is irritated by anything put on the body. The mole should not bleed, ooze, itch, or cause pain. If it continues to be a concern or the mole has changed in appearance, seek advice from a dermatologist.