Do these photos and observations provide enough information to identify the type of paint (Acrylic, Oil, etc.)? If so, what is it? If not, how do I acquire the other information needed?
I'm afraid not, some years ago it would have been a bit easier to guess but these days there are a great many types of paints available (a few distinct types, numerous variations) and they can look identical once dry. And many of them can have a very good gloss level so that's not a clue unfortunately. A good gloss without any surface tack whatsoever in warm weather used to be a pointer to alkyd paint over one of the acrylic formulations (basically any waterbased paint) but they've improved greatly over the years and they rarely, if ever, exhibit this problem.
You may be able to get a good idea what it is yourself by testing the dried paint for solubility. This is essentially the same process that can be used to determine (or at least narrow down) what clear finish might have been used on furniture. Basically you start with the weakest solvent and then move up if there's no reaction, unfortunately this means if you don't get a reaction first time you're looking at buying multiple strong solvents which you may have little or no future use for.
But the good news is I don't think you really need to worry about doing this as the type of paint doesn't really matter for the next step.
then somehow weatherproof the entire table so that it can be kept on a covered patio
This may be your biggest hurdle in fact. The paint may already be quite water-resistant, but I don't know of a good way of determining this on a finished piece without access to the original paint.
You can varnish over paintwork quite successfully to provide additional waterproofing to however much it already has, but varnish can't be relied upon to bond properly to unknown glossy surfaces without thorough surface prep and there are no guarantees of a good outcome.
The usual recommendation to do this is to thoroughly clean, then scuff the surface for the varnish to have something to cling to, which sidesteps creating a chemical bond (this is the ideal where possible) and instead it just clings on physically (less ideal but can still be very good).
The problem here is that the table is white and oil-based varnishes are all slightly yellowish, so you'll end up with a cream table. You can use a waterbased varnish, e.g. a waterbased poly (actually acrylic varnish modified with some additional polyurethane), which are nearly always perfectly clear, or water-white as it's called, once dry. However they are a bit more skittish about bonding well to an unknown substrate.
Regardless of the varnish you use it's advisable to do an adhesion test, taking advantage of the many non-visible spots on the table that you mention in another Question.
So first job is to thoroughly wash the surface of the paint and then rinse off well and let it dry.
Now you scuff it with a fine abrasive. Scotch-Brite or any similar non-woven nylon abrasive pads (including those intended for kitchen use!) work well for this job. Only scuff until you have a uniform dulled surface and then stop.
Then apply the varnish as per the instructions. There's a chance that even with the scuffing the varnish won't apply properly and will bead up on the surface, you can try a commercial flow aid to help with this but it may not entirely solve the problem. If it doesn't regrettably I would recommend you stop and abandon the test as that varnish is sure not to work well enough.
Now the hard part, you have to wait for the varnish to properly dry. If the drying time is minutes and the recoat time is under an hour the full cure period may still be as much as a couple of days. I would recommend you wait at least 48 hours before the next step with a waterbased clear finish. On the chance that you went with an oil-based varnish you should wait at least 5-7 days.
Last step, the test for adhesion. So *gulp* moment, you now score the varnish heavily with a sharp knife in a criss-cross pattern. Score lines approximately 1/8" (3mm) apart, and then apply a good sticky tape and burnish it down well, then pull it smartly off the surface. If no varnish at all comes off you're in great shape, the varnish has bonded exceptionally well. If only a few flakes of varnish are visible on the tape the varnish has bonded quite well and it's safe to go ahead. If large flakes are visible on the tape the bond is poor and if almost all the varnish comes off the bond is obviously almost non-existent.