The grappler just hangs onto the falling character

It's definitely not option (1) because grappling only reduces the Speed attribute of a character to zero, and feather fall's falling movement is not derived from a character's Speed. Even a very, very strict RAW reading doesn't make the grappler and grapplee hang mid-air.

If a weight increase affected the spell it would be noted in the spell's effect, so it's not (3).

I think you've covered all possibilities, so by a process of elimination that leaves option (2): that someone can dangle off a feather falling character. This is also the result of least weirdness, and D&D 5e's rules are written predicated on them being read naturally, with the least weirdness.

I also think this option is fine.

1. As the spell says, the targeted creature's fall rate is reduced.
2. This doesn't prevent them from carrying things while they're falling, but anything, if dropped, isn't suspended by the spell's effect on the targeted creature and would fall as normal.
3. Someone grappling a feather falling character neither breaks the spell, nor is a subject of feather fall themself. Nor does feather fall say that they're magically slippery and can't be held onto.
4. Therefore, the grappler benefits from the reduced rate… at least, so long as they maintain the grapple.

So in other words: go for it, jump onto that falling character and you'll slow down with them, so long as you managed to hang on. Just beware that you're one successful check to break the grapple away from plummeting at full speed away from them.

Yes, but only under very specific circumstances

*Throughout, "you" is assumed to be the creature that is falling attempting to cast the spell on themselves

Unless you prepared an action, you cannot cast earthbind while falling

The rule for falling assumes that a creature immediately drops the entire distance when it falls. 

When you start falling, you drop immediately with no time to take an action. That is what makes feather fall so useful, because you can cast it as a reaction in response to falling which does interrupt the fall. Earthbind takes an action to cast and thus cannot be cast while falling.

The only way to get around this would have been to have prepared an action to cast earthbind if you start falling next round.

Falling from >500 feet (optional rule)

Xanathars Guide to Everything provides an optional rule that allows for non-instant falling.

When you fall from a great height, you instantly descend up to 500 feet. If you’re still falling on your next turn, you descend up to 500 feet at the end of that turn. 

So, if your DM is using the rule and if you fall from >500 feet you would be able to cast earthbind.

If you want to other Q&As dealing with casting while falling look at the answers for Does casting Fly on a creature who is falling cause fall damage? and Can you cast a spell with a Doss lute while falling?

If you find a way to cast earthbind while falling, you (may) still need to fail a save

The target must succeed on a Strength saving throw or its flying speed (if any) is reduced to 0 feet for the spell’s duration. 

Jeremy Crawford says you cannot fail a save intentionally:

No rule lets you opt to fail a save. As DM, I might allow it, assuming you aren't incapacitated or dominated.

So, assuming your DM is abiding by this, you must roll and actually fail a strength save.

If your DM allows failing a save intentionally then this is not an issue.

If you can cast the spell in time and fail the save, you should reach the ground safely

An airborne creature affected by this spell safely descends at 60 feet per round until it reaches the ground or the spell ends.

If you are higher than 600 feet when earthbind is cast then the spell will run out before you reach the ground and you could take damage.