LÖVE

And another...

I have a physics body as the ground, with the player standing on it. How do I detect whether the player is touching the ground, to determine whether he can jump? Or is there a better way?

(At the moment, the player can keep jumping whilst in the air, as if they're flapping wings and flying upwards...)

[Edit] I've kind of hacked a solution to this, by checking the Y position of the player when they press the jump key, and not making him jump if he's too high. It seems quite crap though - is it more standard to do some kind of collision check between the player and the ground, and base it off that?

There are going to be callbacks not only when collision starts, but also when it stops, allowing you to use a simple ground boolean, in the mean time:
You can check if 1 pixel below the lowest point is in the ground, in that case, you're on the ground. (use testPoint on the shape)

Yeah, I've done something similar, cheers.

Q. How do I detect a collision between a bullet and an enemy?

Create the bullet as a physics entity and just fire it, you do need to set identification data and register a collision callback, but the docs of love.physics should help you with that. So basically it's the same as when you were firing other objects.