Microwave door blew open when boiling water-vinegar mixture for cleaning

I presumed the door just blew open rather got blown off and you were referring to the glass that held the mixture rather than the glass window in the door. I am unsure where the soap fits into your cleaning attempt.

With those caveats, unless there is more you have not said, I do not think it was as drastic an event as it appeared. Before I address your questions, it is worth looking into your cleaning attempt.

All the microwave instructions I have read to date (not that many just a dozen or so) suggest wiping the inside walls clean with a damp cloth or with a drop of dishwashing detergent (rinse with water-dampened cloth afterwards). All advised against using any cleaning agent.

When you boil a glass of water-vinegar mixture, you will have some vinegar in the vapours (Raoult's Law) and their condensates on the walls will thus have some acidity. While that may sound iffy, it is actually not outside of normal use. Imagine you have to heat up a sauce with lemon juice or vinegar in it, same outcome. Vinegar is normally 5% w/w acetic acid and you have diluted it further in water (no idea what ratio you used). At such low concentrations, the vapour phase is slightly enriched and you have a slightly more concentrated condensate, still very dilute. (Equilibrium diagram example) So, I would not worry about the effect of a hot and very dilute vinegar condensing or even splashing on the walls, provided that the inner electronics are not wetted. All this splashing and condensing is pretty normal if you were to heat up food inside anyway; imagine if you had to reheat a cup of lemon tea or hot and sour soup.

When microwaving water particularly in a smooth container, there is always a likelihood of superheating (something you clearly already understood) as the water molecules themselves are receiving the energy directly and rapidly unlike conventional heating. Pockets of water would abruptly and periodically find nucleation sites to form vapours and erupt into a full boil with a burst of volume expanion and a sudden spike in pressure inside the oven chamber, sometimes enough to blow open the door even though the chamber is not air-tight. Again, this can happen in normal cooking (too many shell-on and hard-boiled whole egg accidents reported) and I would be very surprised if manufacturers have not already anticipated that.

Now your questions:

1. I would wipe it dry thoroughly and then rinse the inside with a water-dampened cloth and dry it again. Then put in a cup of water and test heat it for say 30 seconds at a time to see if it is behaving normally. It would be best if you could just throw in one or two tea leaves or grains of rice or baking beads to avoid any superheating (this is a good idea normally anyway)
2. Answered in 1 above. If you are really paranoid, leave the door open and air it for an hour. Just bear in mind that when and after you heat food inside, the walls get wet normally.
3. If you are really worried, get an electrician to check it out. Next time you boil a clear liquid, put a few small food-safe particles into it to lessen the chance of superheating.

Aside from that, you might as well try cleaning the stickiness off while you are clearing up the mess. Use a hand-hot water dampened towel if you need to and a bit of diluted vinegar should not hurt either.

I'd be most worried about the door latch and/or interlock being damaged, increasing the risk of microwave exposure if the door came open while running or the microwave didn't cut out when you opened the door. You'd need to test that, and the specifics would depend on the model. Some models where you open the door by simply pulling are likely to be fine, those where you have to push a button are likely to have taken damage to an important part. This type may also suffer a bent dooor, so check for that too.

I'd just clean it out with Formula 409 as usual, a scrub and rinse with water, towel dry, and see if it still boils a coffee cup of water in a minute or two. If so, you should be good to go. I don't like using vinegar to clean my microwave as there are microswitches and electronic connections that do not like being exposed to corrosive acids. Not a big problem, but it could be a long term issue. The safety interlocks can be tested by opening the oven while it is running. The oven should stop running when you open the door. Sounds like you had quite a nice size superheating episode there!