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.
Tracing is the first thing I would try.
If the trim detail is not super fine, sand lightly to remove burrs and bumps, and then tape (painter's tape probably, or whatever you have on hand that is close) along the the path the pencil will take to bridge valleys and pits.
Best Answer
They make several styles of tools that are made specifically for getting nails out. One tool called a "cats paw" looks like the picture below. (You may find similar tool under other localized names). A hammer is used to pound the hooked end in under the nail head so that it can then be levered out.
Another type of nail puller is also very effective in that the tool is used by itself without a need for a separate hammer. The handle on this type of nail puller is used to slide up and pound dowwards to embed the jaws around the nail head. Pulling sideways in the direction of the "foot" forces the jaws to clamp tightly on the nail and pull it out.