Learn English – Fall back in chair / fall back into chair

Prepositions don't have to be part of a phrasal verb, in fact I would say most aren't. In this case, none of the prepositions have any special relation to the verb. They're all literal descriptors of the positions the subject goes through during the action.

Try to break up the sentence into the smallest chunks that make sense. Here: He fell. Yeah, that makes sense, that's true. Okay, where did he fall? Into the pool. What did he fall off of? He fell off the ledge. So now, put it together.

He fell off the ledge and into the pool.

You can leave out the "and" here, but it's still there for grammatical purposes. You can tell if you use three prepositions:

He fell off the ledge, through the air, and into the pool.

You can't leave out the "and" here, there's no way to use three prepositions without the conjunction. That means that there's no way to do it with two, either, and the "and" is still there, it's just not spoken.

Right and straight are both intensifiers, and don't really make sense to me here. There's not exactly an indirect way to fall off a ledge and into a pool.

I hope this helps.

Edit: A note to help distinguish between phrasal verbs and normal prepositional adverb phrases: phrasal verbs don't accept objects of the preposition, and they don't pair with prepositions in conjunction phrases.

The roof fell in. good
The roof fell in the house. not good
The roof fell in and onto the ground. not good
The roof fell in onto the ground. good

Does that help you see the difference a little better?

Why usually is the word “halfway” used with “down” rather than “up”?

I doubted that. I can't imagine that there is some kind of rule to use down rather than up. It's possible, but I didn't see one. I looked to Ngram.

So the usage looks roughly even, at least in recent years.

One reason you might use one over the other is perspective. For example, in a larger context (like a story), if you want your audience to visualize themselves looking down at the nest (fixed, not moving), then you might use halfway down. If you want your audience to visualize themselves looking up (at the base of the cliff) at the nest, then you might use halfway up.

In a slightly different use, imagine a group that is at the top of a cliff and they have a friend climbing up. He reaches the halfway point and one of the friends reports "He's halfway up the cliff." In this case, halfway up implies that he is moving and he is moving up. You can construct a similar case for halfway down.

The given exercise/example is so brief that I don't really see a difference. It's possible that the author was "looking down" or wanted the reader to look down, but I think either choice, down or up, is equally likely and acceptable.