Let's assume that a monster shares a space with an investigator and there is another investigator on an adjacent space and another investigator 3 or 4 spaces away.
The app says "the monster moves 2 spaces towards the nearest investigator and then it attacks any investigator on its space".
I assume that the nearest investigator is the one that shares the same space with the monster. Does that mean that the monster does not move at all? Or does it move 2 spaces into the space with no investigator? Or does it move one space to the space occupied by the adjacent investigator?
As mentioned, the app says "moves 2 spaces" and not "up to 2 spaces".

Same question in case there was no investigator an adjacent space. Does the monster move 2 spaces towards the investigator in range 3 spaces away??

An since it does not state "in range" does it move through doors to get the to investigators away from it?

In your example the monster would not move. The key is it says towards the nearest investigator. You check to see which investigator is the closest in the current board state. In this case it is the investigator already sharing the same space, therefore it doesn't need to move because the requirement is met.

In your second question regarding range, the game will specifically state when to consider it. In your example you would simply move the monster two spaces towards the closest investigator that is three spaces away. Since range isn't specified you would count through doors or any other legal movement that monster is allowed. What would likely happen is when that monster is a space away it will tell you to move closer since it has no one to attack at the end of that first command.