Spells do what they say they do; lightning lure does not say that it grants any sort of flying or hovering ability to the targeted creature. It is a lash of lightning energy, so it is physically pulling them(the strength save). It is not a tractor beam so much as a whip.
If using basic falling rules it is up to interpretation...
... but most GMs ...
Yes you can pull the creature
Lightning Lure says:
The target must succeed on a Strength saving throw or be pulled up to 10 feet in a straight line toward you
As opposed to this spell, we can consider the 1st level spell Dissonant Whispers. The spell's description says the following:
The creature doesn’t move into obviously dangerous ground, such as a ...
The spell description says only that target is being "pulled", without further clarifications of the process.
However, Lightning Lure asks for a Strength saving throw. This implies the target can use its pure muscular strength to resist being pulled, so it probably stays on foot during the spell effect. If the target was afloat during ...
Ask your DM
This is not explicitly specified in the description of the spell, or the rules. It will depend on how your DM interpret the spell description if you do a 10-ft pull and a hole is between the caster and target.
What I would do: resolve the spell as up to 10-ft pull, regardless of terrain. If after resolving the target ends up on a tile with a hole,...
If you want to extend a jump, you need an Athletics roll.*
p175 gives this as an example of an athletics roll.
You try to jump an unusually long distance
So, a cliff would serve as a way to argue to a DM you should go further and have it be easier. How much further, and at what DC and whether you get advantage is up to them.
You do not gain any additional jump distance
The rules on Long Jumps state:
When you make a long jump, you cover a number of feet up to your Strength score if you move at least 10 feet on foot immediately before the jump. When you make a standing long jump, you can leap only half that distance. Either way, each foot you clear on the jump costs a foot of ...
Falling damage on its own isn't a thing, you take damage from fumbled skill rolls.
Fall damage is a reflex, athletics thing that happens when you fail at climbing. See p50.
Athletics. This skill is required for accurate throwing, climbing, and balancing.
p43 covers the consequences for falling on the fumble table, which range from nothing happening, to 1 ...