What is a reasonable occurrence of dungeons in a typical Medieval fantasy world?
In theory you could have several dungeons (as defined in the question) on top of each other or only a few in the whole world. It depends on climate and sources of food.
If we look at <PERSON> text using late-medieval France as a default, it is suggested that "square mile of settled land (including requisite roads, villages and towns, as well as crops and pastureland) will support 180 people". The text also mentions that not all land is suitable (for settlement that is) making the average density for bigger areas considerable less, and (as pointed out in comments) "The French were blessed with an abundance of arable countryside".
Anyhow, if the dungeons inhabitants are living of raiding the surrounding land, the number of non-raiders will after a while (or in worst case: immediately) be lower to account for the resources taken by raiders. When enough resources is taken by raiders, the non-raiding population can not sustain itself, or will get fed up and do something about the raider-problem.
But what if the dungeon is self-sustaining? Well in that case we could have "infinite dungeons" (as long as we can build a "deeper" cave), but will need some sort of magical or non-magical energy/food source. The amount of farmland needed per person depends on circumstances. This becomes even more complicated when we take into account that the farming will take place in caves, that might or might not have magical radiation, one would have to imagine that the plants chosen does not use photosynthesis (such as mushrooms), rain does not fall directly on the crop and so on.
Regarding dungeons that consist of ancient ruins, I imagine that these are populated by the undead or wild animals, but once where populated by humanoids. It does not have the same need for food supply, but rather a need for explanation. Maybe the ancient ruin got caught in a earthquake/curse and the city moved a few hills over, or maybe the whole ancient civilisation collapsed and places of lesser importance in the relative post-apokalyps become abounded and relatively forgotten (imagine Machu Picchu). If all ruins are meant to come from the same lost civilisation one could model the city density of the civilisation to get the ruin density. 878
If a ruin is close to active settlement, there must be some reason for people not going over to dismantle the ruins when they need materials. Such reason as the undead, curses or difficult terrain.
What is a reasonable amount of time for a dungeon to "repopulate"?
Again, in theory it could take days or they could never repopulate. It depends on how attractive the living space is and how dense the population is in the surrounding area.
A raider cave could be repopulated in a matter of weeks or even days, as long as there is beings that want to raid that can catch the rumour that the "district" close by (or further away if directed by a Dark Overlord) is empty. If they bring some resources (and gain some on the way) it would not take long to settle in.
Similar for self-sustaining dungeon that is part of an cave ecosystem. Maybe the surrounding tribes of similar humanoids used to trade with the affected dungeon for a certain type of mushroom, and now find themselves with opportunity to take over the whole mushroom cave.
But on other hand maybe that particular cave had a bad reputation, the raiders where the only ones desperate enough to live the raiding life and/or the dungeon have nothing more desirable left except the living space itself. Then it is probable that, aside from a few wild animals, the dungeon will be left alone to the ghosts.
If we are going by repopulation by population growth alone, it could take a long time. Medieval population growth is affected by harsh conditions such as famine, and fatal illness. High birth rates and high death rates, gave a small increase over time. If the dungeon-dwellers have a faster reproduction rate than humans, population growth will be more viable. 121