Depends on what the table is used for. Given the description of wear, let's assume it's a college project work table, that needs to handle anything from homework writing, gaming, to x-acto knife cutting, to light woodworking, painting, etc.
I would suggest hardboard (aka Masonite). It ain't fancy, but it's cheap, widely available in 4 ft x 8 ft sheets (so no cutting required), and suitable for this level of usage.
You could glue it to the existing surface using regular wood glue, however you might think about instead just nailing it in using a lot of small tack nails; this way if the surface becomes too heavily marred, future maintainers could just rip it off and replace it with a new sheet of hardboard.
I use hardboard for surfacing my work tables in my shop. I generally throw a couple coats of polyurethane on top. I suspect you could use pretty much any sealant you wanted on it (or none!) If it's going to be used more for homework/gaming, and less for crafts, a hard coat might make sense.
It depends on whether the wood is actually singed or just the finish.
You need to remove the finish on the door (and perhaps the whole cabinet unit), but begin with a test stripping of just the damaged area. This can be done with solvent strippers (harsh and smelly, but faster) or citrus based (a bit slower but more person/environment friendly). The wood then needs to be lightly sanded.
Refinish
If the wood is not charred (black or severly discolored), you can strip the rest of the door, sand it, stain it to match the rest of the cabintet and finish it with a polyurethane finish with a sheen similar to the cabinet. Getting it all to match can be hard, so you may wish to do the entire cabinet.
Repair and Refinish
I the wood has a slight char or discolorization, you can try to sand these areas out or use a diluted wood bleach to match the rest of the wood. If this works, then proceed as with refinishing above.
Paint
If the damage is too deep, you need to cover it with an opaque finish, such as paint or a heavily pigmented stain. This will change the overall look so it needs to be done to the entire cabinet. Sand away the damaged finish and lightly sand the rest of the piece with 120 grade sandpaper. Then refinish according to the paint/stain directions.
Countertop
If the laminate is blistered, there is little that can be done except replace it. If it has just lifted, you may be able to reglue it. This generally will not work unless the lifted section is near an edge and can easily be pushed flat. If you can lift the whole section at least 1/2 inch, contact cement works best. Spread glue on both faces to be joined and let dry separately before pressing together. If the gap is too small, you can force conventional glues under the area and weigh it down while the glue sets.
Best Answer
The use of distinct solid-wood edging strongly suggests veneer
"Bookmatching" (mirrored grain) is not a guarantee of that, as it can be done in solid wood, but the close match of 5 pieces strongly suggests veneer. So it may just be veneered both sides. When solid wood is "multi-part bookmatched" the thickness of the wood (and change of grain in that thickness) means the match on faces not simply "cut and opened" is not close. To have that close of a match suggests thin slices - veneer.