Well, this case certainly is open to some interpretation, so it's really up to the DM to make the rulings.
- Does the water push them down faster than Feather Fall allows? (60'/rnd)
The spell basically sets a creatures base falling speed. Since it doesn't specify that that speed cannot be changed by other means, it could theoretically be influenced by e.g. another spell.
However, afaik there is no effect that specifically increases fall speed.
Whether water does this is up to the DM.
My recommendation would be no, as otherwise e.g. heavy armor or carrying another creature could make the creature fall faster and I assume that goes against RAI (Rules as Intended).
- Do they take damage from bouncing off of rocks, or do we assume that moving at just 60'/rnd, the damage is mostly avoided?
Well 60'/6s is a speed of 10.9728 km/h (6.8181 mph).
As a comparison: regular walking speed is around 5 km/h.
If you walk against a rock, I doubt that would do enough damage to justify actual D&D damage and even with twice the speed it would be negligible damage if even that.
The other part is whether that would be considered landing (again I'd say no, otherwise if touching rocks is considered landing, is touching a bird or an arrow that got shot at you also landing?). Again, this is completely up to the DM and both rulings would be acceptable.
- Do they tumble around in the air due to the water pushing them? ie. it would be difficult to shoot range weapons or cast spells?
Feather fall does not specify that it stabilizes them.
So depending on the terrain and how controlled they entered their fall, you might wanna impose disadvantage as a separate ruling.
Feather fall itself does not mention any disadvantage and underwater combat does so only with specific weapons (if you consider them fully submerged in the first place).
- Are there any other relevant factors to this scenario that I'm not thinking of?
This is a edge case and doesn't really have RAW and as such, a DM is expected to just make an impromptu ruling on the situation anyways.
My first thought at an answer is that the water and player are both falling (vs. the water pushing them), so they "float", and when the player hits a rocky outcropping, it is considered "landing" (no damage) at which point they aren't falling anymore and the water immediately pushes them off, at which point they are falling once more. I'm not sure about tumbling around.
This is a completely acceptable ruling and I wouldn't object to it as a player.
As a DM, however, I, personally, prefer to rule in favor of the players, if in doubt.
In the grand scheme, whether the player gets more fall damage or not, doesn't really matter*.
However, deciding in the favor of the players makes them feel like the successfully navigated an obstacle and used their spells smartly.
While letting them fall the rest of the way might give them a feel-bad effect, where they thought they were creative and had a good idea, but feel punished for it, since it didn't work out and they uselessly used some resources to cast Feather fall.
This can be rather demotivating.
*) Unless they die from it, in which case the feel-bad effect is even stronger.