I'd really need to see a photo to know, but more or less...
A lot of your options aren't great for food prep surfaces: for a non-food surface, I'd use turpentine or Restor-a-Finish.
Start with baking soda, warm water and a green scrubber to get the gumminess up. You might try washing soda or dish detergent if the baking soda alone doesn't do it. Fill a small bucket with warm water, sprinkle some baking soda down, scrub it up, wipe it up with a wet dishcloth, rinse the dish cloth, repeat. I've heard you can alternately use white vinegar. You probably won't scratch the wood but it never hurts to go with the grain instead of in circles.
Either way, don't leave standing water on the counter, clean it up as you go.
If it is really nasty you might wind up with a few rinses. Once you get it clean, oil it with mineral oil or refinish it. If you go the mineral oil route, you pour a small amount on (small) and rub it in (with the grain) and repeat.
UPDATE: I just happened upon a blog post which suggests that a bowl of hot water, a few tablespoons of murphy's oil soap and a pad of steel wool will go along way. With steelwool and sand paper you definitely want to go with the grain, always. Never ever go in circles. Add the soap to your hot water, scrub with the grain, rinsing often as you go. Change the water as it gets grimy. Keep some rags handy to dry up with so you aren't leaving standing water.
As far as the warping goes, you probably can't do much except plane it, is a recipe for making everything worse if you don't know what you're doing. Are you moving soon or just starting to realize you maybe need to be more conscientious?
To add stability you could use 2x6 members for the cross pieces along the bottom. Then cover the bottom with a half of a sheet of plywood. Pour concrete into the box thus formed at the base by the 2x6's and plywood.
You could get by without having to go all the way to full angle braces across the sides (which would get in the way of your hanging plants anyway) by installing some gussets made of thin plywood as shown below:
At a suitable size these would probably come right up to where the first hanging pot wires go.
I see that you have each corner made of just one 2x4. I would encourage you to make the corners out of two 2x4's attached in an L shape in each corner. This will go a long ways toward making the unit a good bit sturdier and will also make the cross bars on two of the sides have a whole lot more wood to bolt into.
I would echo that the proper fasteners for this structure are going to be 1/4" or 5/16" diameter carriage bolts as opposed to screws or nails.
Best Answer
The tiny black specks that are not knots in your pictures are most likely from small particles of steel or iron causing staining. I wonder if your refinish process used steel wool (not a good idea on oak unless you are trying to stain it black.) The larger features with actual holes are, of course, knots, so if you have "holes that are not knots" you have not shown any pictures of them.
That new picture is a bit hard to see detail in (as you know) but there's certainly something knot-loke going on on the grain around it, so it probably is as well. Sometimes on an older floor that's been sanded down a lot you start to see stains from the steel nails, but that seems less than likely here.
To skip right to your water concern - probably not, since there's no sign of the floor buckling or failing to lay flat - wet hardwood floors tend to look like #expletive omitted# in short order as the bottom swells and the boards cup.