I am trying to understand whether the Curse of the Sea feature of the Sea Sorcerer subclass (from Unearthed Arcana: Sorcerer) works with the warlock's Repelling Blast eldritch invocation.

The Curse of the Sea feature states, among other things:

Once per turn when you cast a spell, you can trigger the curse if that spell deals cold or lightning damage to the cursed target or forces it to move.

The Repelling Blast invocation states:

When you hit a creature with eldritch blast, you can push the creature up to 10 feet away from you in a straight line.

If I'm going to be multiclassed in Warlock and Sea Sorcerer, can I trigger the curse off of eldritch blast with Repelling Blast, or not?

It sounds to me like Repelling Blast is a power your character has, not an add-on to the spell, so I tend towards "no," but it kind of feels like they should work together.

It's going to be up to your DM

This is definitely an edge-case - and even moreso because it's language used in Unearthed Arcana which Jeremy Crawford has stated that:

Unearthed Arcana material isn't tuned for multiclassing. We make a class/subclass multiclass-ready if it's going to become official.

Odd interactions like your question are bound to come up and will likely be dependent on tables as to how to adjudicate if multiclassing is pursued.

A reasonable approach

Having said that, it does seem like the two abilities should interact. The requirement for the curse is movement due to casting a spell, which is exactly what is happening with Eldritch Blast with Repelling Blast.

Your DM may initially allow it because of...logic :) But table experience may determine if the combo is overpowered or not. If so, don't be discouraged if they want to roll back the allowance and bar the interaction.

Multiple Blasts Concern

As Ben Barden pointed out, EB provides more than one blast. No other sorcerer cantrip that delivers cold, lightning, or forced movement does this. This is another interaction unplanned with the UA multiclassing.

Once again, it's up to the DM how they want to handle the curse application here. But given a successful series of hits could then push the target back in multiples of 15, this suggests a problem with the interaction and a reason not to allow it. However, if they decide "that's awesome!" and it works for them and the table - then allowing it is reasonable as well :)

Yes. It is stated clearly in the description of the ability.

Once per turn when you cast a spell, you can trigger the curse if that spell deals cold or lightning damage to the cursed target or forces it to move.

So if the Storm Sorcerer casts a spell, any spell, that forces target to move you can trigger the curse. If the sorcerer casts Eldritch blast with Repelling Blast then the creature is forced to move by the spell and the curse can be triggered.

As another example, if you had an ability that allowed you to substitute one elemental damage type for another (there is one somewhere, I just can't remember where) and you made your fireblast cantrip do cold damage, that would also allow you to trigger the curse.

As to the multiple rays, they do not happen simultaneously, as Jeremy Crawford rules here, so one ray could hit and trigger the curse and then a second ray hit the same creature following the curse taking effect.