1) Yes, one object can apply force to another without even touching it. For instance, it is called magnetic and electrostatic force, electrostatic and magnetic force is a different type of force that can pull or push another object without having to touch them together. For example, think of a magnet, if you put the North Pole side of the magnet and the South Pole side of the magnet they will pull each other together because of the different electric charges. But the two north poles will push each other away, like others say they repel each other.
2) A magnet does not have to be touching an object to attract it or repel it. This works because magnetism can work over a distance. Even though we can't actually see magnetic force, we can see what the force does to things around it.
I hope this helps! If not I’m sincerely sorry! Have a good day/night.