Yes, it is called "our judgement".

If the ball handler has the rights to the vertical space above the player, is it illegal for him to use the hand to block/push the defender's hand/arm away should the defender extend his arm into the handler's vertical space during layup, assuming the ball handler's arm remains within his vertical space and not extended into the defender's vertical space.

Also for a defender trying to block a standing still handler trying to pass the ball, is it illegal for the defender to extend his arms position towards the handler to restrict the handler's arm movement (close but no contact was made however the handler won't be able to make a pass/turn/swing his arms without coming in contact with the defender's arm/chest and even so his movement would be restricted). Would you call it impeding of movement on the defense when the handler arms comes into contact with the defender and arm movement restricted?

If you put your hands/arms into an opponent's vertical space, the onus is on you if there's contact. So for your second example, if B1 is playing defense with his/her arms reaching towards A1 and A1 moves his/her arms into B1's arms there really can't be a foul called on A1 (assuming A1 doesn't karate chop B1 or something like that).

If you are going to post on a basketball official's website you need to ingrain it into your mind that there are no personal fouls if there is no contact.

I understand the no contact = no foul thing, but my question is related to contact made by A1 (ball handler) because D1 was putting his hands around A1 with no contact but A1 has to initiate contact in order to pass his ball because he's surrounded by D1, would A1 be called for initiating the contact or will D1 be called for impeding of movement?

So meaning A1 has the right to defend his verticality with his body/arm,

what if:

A1 did not extend his arms into B1's vertical however pushed B1's extended arm away (within A1's verticality) and makes B1 fall down or cause B1 to be displaced while on air?

A1 initiate contacts with B1's extended arm within A1's verticality, would B1 be called for a defensive foul? For example B1 extended his arm towards A1 A1 raises arm to block B1 continues pressing his arm towards A1's blocking hand.

As I'm seeing these plays in my mind, both are either fouls on A or no calls.

If A1 is using their arm to push B1's arm out of the way, then verticality doesn't come into play. There is never a case where a player is permitted to use their arm to push any part of another player.