[RPG] use Lightning Lure as a grappling hook

Does lightning lure pull a creature toward you regardless of other circumstances?

No. The key words here "up to 10 feet" If something prevents it from being pulled, that would pull it 0 feet, which is included in up to 10 feet. Likewise, if there is suddenly a wall 5 feet away from the creature between you and it, it would stop when it hits the stone wall. They still roll the save, and on fail they move 0 feet but if they are within the distance, they still take the damage. That is my interpretation of the RAW.

I'd argue that grappled is muddier and that would require a DM call. The question is what is stronger the grappler's hold on the target or the spell. I'd personally rule that when the target makes a failed save, a contested roll be made between the spell caster's spell attack modifier and the grapperler's strength roll.

No.

Here are the relevant phrases in support of my answer, from the rules for Opportunity Attacks.

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.

You can avoid provoking an opportunity attack by taking the Disengage action. You also don't provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.

1) Lightning Lure does not result in the target moving of their own volition. It moves them without using their movement, action or reaction, and a standard Opportunity Attack only occurs if a target moves of their own volition (or while under the effect of some charm or other condition that causes them to move by coercion as if they had done so willingly).

2) Polearm Master only explicitly extends the general rule that Opportunity Attacks happen when you leave a creature's reach with a specific extension that it can happen when you enter a creature's reach. The contrast of the words "leave" and "enter" implies that Polearm Master is only intended to alter the direction of intentional movement on the part of the target that triggers the Opportunity Attack, not other aspects of how Opportunity Attacks work. If that were not so, we would expect the Polearm Master description to state a special exception to the general rule by allowing this Opportunity Attack to trigger against a target that is unwillingly moved, but in the absence of such an exception the prohibition on unwilling movement in point 1 still applies.