Yes friction of any kind (kinetic or static) always opposes motion. Think of it is way, when you have a piece of sandpaper and rub it against the wall to remove paint does it matter which way you go? No, it doesn't matter which way you move it the paint in that direction will be removed.
For ice skating. no the friction will always work against you. Think of a hockey puck on ice, does it get accelerate (continue to get faster) after a soft push? No, it eventually slows down and comes to a stop. Ice skates just have a smaller point of contact with the ice than hockey pucks, but the principle is the same.
Static friction simply acts to keep the object still. If you are trying to push a heavy box on a flat road do you gain an advantage by pushing on one side over the other (assuming you are pushing in a direction parallel to the road)? No, it's going to take the same amount of effort whichever side you're pushing on. What you're confused about is the normal force. The normal force is the force that acts to keep the box from falling through the floor and to the center of the Earth. It's the force that floor is putting on the box. The static friction force is the normal force (mg if the object is on a flat surface, or mgcos(theta) if the object is on an inclined plane) times the coefficient of state friction (mu_s)
Here's a Khan Academy video if you want a step by step