How often per round can the ability of a pet/Artifact be used? Is there a differentiation between abilities whether they are once per round or “Passive” buffs and how to differentiate?

The rulebook clearly states that items and pets can only be activated once per turn. If it is a passive ability it is considered “always on” and does not need to be activated. If something does something like exploring or placing a token that is an activation. If it says “Ignore walls” then its passive.

What exactly can be stolen? In the rules it says you can’t steal cards, but then there are cards that let you steal items or pets. Are those exceptions or are items and pets played beside your board aren’t considered “cards” for the time being until they are back in your hand? Apart from that I assume unless specificed in the scenario, everything but cards are considered things that can be stolen?

The steal effect refers specifically to stealing carried collectibles ONLY. This means crystals, sheep etc. Cards in-hand cannot be stolen except through the use of cards or abilities that specifically target cards in hand. Cards like rune of finding specifically allow stealing of items in pets IN PLAY from a specific range.