If the surface is edge grain and the previous finish was only wax, wiping down with mineral spirits and then sanding using progressively finer grits up to 220 or so should be pretty good preparation to refinish. Skip the soap and water treatment as it won't dissolve any wax and only complicates finishing. Before applying any finish the wood should be smooth and dry. Any staining or blotching you can see is not going to be hidden too much by a clear finish - it's sometimes possible to bleach out stains but this is tricky to do well.
If you are still getting something soaking back up after sanding down to the raw wood, it could be a prior mineral oil treatment. Mineral oil never hardens and is often used on wooden butcher blocks and countertops. Simply reapplying more would be one refinishing option. It's easy (you just apply until it is no longer absorbed) but you have to reapply it frequently. Inexpensive pure mineral oil can be found in a pharmacy (it's sold as a laxative).
Another option would be a pure tung oil finish. It would be slightly more expensive but a little more water resistant. You can get it from a woodworking or finishing supplier. Most "tung oil finishes" sold by hardware stores are not actually tung oil so check the labels carefully if you go this route. It would be applied and reapplied similarly to mineral oil.
Some "Danish oil" finishes are a blend of varnish, oil, and thinner, and others contain only thinned varnish. Generally one applies "Danish oil" in light coats such that a film is not really built up on the surface of the wood. Often wax is then applied over it. This feels nice on furniture but it is going to be less durable that a slightly thicker application of regular varnish applied with a brush or thinned and then wiped on in several coats.
The mineral oil or tung oil would be my choices for a counter that you are actually going to cut on. On the other hand if you don't plan to cut with knives directly on the counter, and if you are able to prepare the counter down to a clean, dry, smooth surface, a wiping varnish is probably the most durable and best looking option.
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.
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Keep your eyes open for newspaper or online notices of Yard Sales & Estate Sales, these will also include Landlords, Stores, Hotels & Casinos looking to get their buildings cleaned out by the public wanting to remove anything.
Constantly check Craigslist types of websites, your phonebook & internet for Auction, Salvage & Reclamation companies & yards. They usually get the best stuff because they're on the call list of Demolition companies.
Finally, check Salvation Army, Goodwill, Green Drop & Habitat For Humanity. And, consider dismantling a large old Bedroom Dresser or Armoire or an old Diningroom Table or maybe a China Cabinet & an old Livingroom Coffee Table.