1: Yes, they may activate the glyph. I believe there was a ruling in the most recent FAQ that stated that if a monster is on a glyph that is activated, it must move off as soon as it is able to (i.e. during the monster's normal activation).
2: Yes, collecting coins happens as soon as they are on the pile.
3: No, you may not open a chest unless you are the only thing on it. Pg 18 of the rulebook states, "A hero cannot open a chest if the chest is in the same space as another figure."
I agree with Shnar except that collecting coins does not happen 'automatically'. It is a movement action that costs 0MP and cannot be done until the hero is in that space. If there was some reason that the hero could not do a movement action (like being a monkey) or some effect which killed the hero as it entered the space (like Aura), then the coins would not be collected.