I used a tall plastic bucket once to catch the mice in the garage - I just put a small amount of dog food in the bottom, they could not climb the walls or jump that high to get out. I used a "ramp" made of simple cardboard to the top so that they could hop right in! Safe, no poison and easy to carry out when "full".
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?
Best Answer
Code requires fire retardant foam in new construction nowadays. If you have an older home where that was not required, you could leave it or fill it with standard caulk, if you feel you must do something.