You bet they can climb, not like squirrels but pretty close. They can also climb a 5 foot chicken wire fence, get inside and kill chickens. Skunks are opportunistic, they like to eat just about anything that is offered them. Garbage and pet food left out are two of their favorites. Their natural diet is insects like roaches, grub worms in you lawn etc. When these foods are no longer there they will move on to tastier locations, but not if anything is left that they like, UNLESS they are actually living there. They love to live under elevated buildings, in walls of sheds and in old abandoned cars. Whenever I have had skunk problems I found out later that they were actually living very close by. I found out by accident that they absolutely hate the smell of oranges/citrus. I was killing fire ants with orange oil (d-limonene) and was pouring the stuff as a drench over ant mounds which were everywhere. I guess it became too much of an odor because a momma skunk and her 4 little one scampered off out of the area before it became dark outside and unfortunately my two dogs spotted them and you can guess the rest of that story. Ever since that memorable day whenever I see little dug up areas in my garden or lawn I know it is either skunks or armadillos and armadillos make larger and more numerous holes than do skunks. I bring out the orange oil and spray everything and everywhere they might be but just keeping the area devoid of anything to eat is the best remedy for them.
Almost every house has some sort of citrus in it.