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Catapult reads:

Choose one object weighing 1 to 5 pounds within range that isn't being
worn or carried. The object flies in a straight line up to 90 feet in
a direction you choose before falling to the ground, stopping early if
it impacts against a solid surface.

Could you use this to fire objects directly upwards or at an angle (like a real catapult)?

If you can fire at an angle, can you use that to extend the range? For instance, traveling 90 feet diagonally then beginning its projectile descent. I'd assume accuracy would be much more difficult than the straight-line 90 feet.

What about using it to launch grappling hooks, etc., with ropes for climbing purposes?

Can you fire at an angle? Yes. Does it extend the range? No.

Catapult does not specify that the object flies horizontally. Nor are there any additional rulings from Sage Advice on the matter. Pick a direction and watch it fly.

But no matter what direction you point it, the range of Catapult is 90 feet, at which point it "falls to the ground". While it does not specify whether it full-stops or keeps going, the spell damage only applies to the 90 foot range.

A GM can homebrew that it would continue to sail and roll, but it would use a different damage system, no longer empowered by the spell. A good reference for rolling damage would be the Sphere of Crushing Doom trap from the trap UA, which is a 10 foot wide steel ball that rolls downhill, 150 feet in 6 seconds, and does 4d10 damage on impact with a creature. (Obviously a 10ft steel ball weighs a lot more than anything you could catapult, around 500lbs per cubic foot, so keep that in mind).

There's also the question of damage while the object is falling. This question has an answer that includes a GM-made nifty little chart a GM could use for falling damage.

In the case of a grappling hook or utility use, I'd probably put it to your DM whether it hard stops at 90 feet or not.

Yes, you can choose any direction, including straight up.

As the spell describes it, the motion of the object is very strange. It moves in a straight line, so it's not a projectile in the usual sense. It's more like a rocket, where the effect of gravity is counteracted constantly during its flight.

It's not clear whether the object continues its forward motion.

The object travels 90 feet "before falling to the ground". Does that mean that mundane physics reasserts itself and the object immediately starts falling like a projectile, or does it fall straight down? From the text of the spell, it's not clear.

The first reading has the problem that it's not clear how fast the object moves, so we as players can't really estimate where it will end up. This is one of those situations where as a DM you have to consider what the player is trying to do and the likely results of success or failure, and then rule on how to resolve it.

If the latter, then launching the object at an angle isn't very helpful, as it just shortens the range. You still might do it to hit a target in the air, or launch a grappling hook or something, as you suggested.