1. MDF core table is probably better over time that a solid wood table. It has it's downsides, like don't get it wet, but overall it resists warping and shrinking much better. Real wood can move a ton! There are more reasons plywood is in use than cost. In fact, I can make a 4' x 8' x 3/4" sheet of solid knotty pine cheaper than I can buy a good quality knotty pine sheet of plywood, and in almost every instance I will prefer the plywood.
2. As far as a "real" goes, that table looks like it could be a fairly expensive cherry or mahogany veneer table. Hard to tell due to the angle of the photos, and the fact that most stuff sold as cherry is actually poplar, and most mahogany sold isn't at all. Unfortunately corners were cut in its manufacture. MDF should never be left raw even if it's hidden. It's essentially very thick paper, and as such easily takes on water from any moisture.
3. The table is fixable. The swelling is higher than the veneer is thick, so a simple refinish won't do because you will sand through the veneer, but one MAY be required anyway. If I had to fix it: I would take the top off the table. I would use a router to cut about a 1/4" wide slot, 1-1/2" - 2" deep in the edge of the MDF under where the swelling is. I would then secure the table top to the side of a workbench vertically, with the slot facing up. I would then fill the slot with SLOW CURE epoxy resin, leaving it just slightly under-filled. West Systems 408 or 403 is going to be the best resin for this project. I would have two pieces of 3/4" MDF or melamine, two clamp cauls, about 4' x 8"-10", 3-4 12" deep-throat bar or clamps or large deep throat C-clamps, and a box of waxed paper ready to go. Use blue tape to tape the waxed paper to the top of the table, but don't get any tape where the clamping boards will go. This will protect the top from scratches and leaking resin. Then I would place the MDF on each side of the slot and clamp it VERY CAREFULLY and with EVEN PRESSURE until the bulges are flush with the table top, and then tighten it just a smidgen more. Too tight and you will leave clamp marks in the tabletop, even through the MDF. I highly recommend cauls on top of the MDF to even out the pressure. The slot will allow the clamped down bulges a place to go, and when the resin hardens it should act as a filler and glue, holding the table top to uniform thickness. Some of the resin may leak out when you clamp and the slot gets narrower. The ideal is to have a slight convex meniscus along the length of the slot. When the resin hardens it will shrink some, and you can easily trim of the excess after it hardens.
This may leave the table top with cracks in the veneer where it bulged, in which case the top will need to be stripped and refinished.
If the table is under $1000 I wouldn't bother.