Epoxy. Gorilla Glue is exterior rated, but makes a mess, a big mess if you use too much as most people do until they get used to it. Your instinct to avoid nails or screws is good. It concentrates stresses and almost always does more harm than good.

That's tough as the veneer has lifted. If you sand it down it will look worse. I suppose you could use a thick poly pour on top but you will still the lifting beneath the pour. Quite frankly I cannot think of anything else you could do other than laminate a new top over the one you have and then it might not match the chairs. You might consult a local furniture repair company and see if they have some suggestions. The isue is how much money do you want to spend working with the table.

Burns can only be sanded out. With a veneer like that, they may be all the way through it. Use a dual action sander to keep from sanding divots into it and give it a go. The worst that can happen is that the burns do go all the way through. That would be an opportunity to cut the veneer out and maybe do a decorative inlay with a different veneer, or learn how to do a patch.

In the roughly 10,000 pieces of furniture that I've repaired, there have been 3 that were not some sort of lacquer. Highly unlikely that any commercial furniture made after 1940 is shellac. The reason Pledge was noted was that its silicone oil will dramatically affect most finishes with a problem called "fish-eye" -- little craters when a finish is applied over the top of it. The lifting of the finish is most likely due to a crack in the finish that has allowed water to get underneath and compromise the finish. Letting it go is only get worse.. While there are lots of repair approaches depending upon materials and skill available, you've spent a lot of money on this table and I'd suggest not trying to do a DIY and really messing it up. Contact the seller and see if this is a warranted claim damage. If not, try to find a competent furniture repair technician. A repair might include excavating and inserting a "dutchman", stripping and refinishing the top, filling with polyester filler, epoxy, burn in sticks, or really thick "repair lacquer" depending upon closer evaluation.