Bevor ich ins Bett gehe, falte ich das Handtuch zusammen.

Because bevor introduces a subordinate (dependent) clause. In German, subordinate clauses usually send the conjugated verb to the end. So you get … ich ins Bett gehe (literally: “... I to bed go”).

They belong to two different clauses, each with its own conjugated verb:

• Subordinate clause: Bevor ich ins Bett gehe → verb is gehe
• Main clause: falte ich das Handtuch zusammen → verb is falte

Because the sentence starts with the subordinate clause, and that subordinate clause takes the first position in the overall sentence. In German main clauses, the conjugated verb must be in position 2 (V2 rule).
So after the initial clause, the verb comes next:
Bevor …, falte ich … (not …, ich falte …)

ins is a contraction of in + das:

• in das → ins German commonly contracts in das, an das, auf das, etc. in everyday usage.

ins Bett uses in + accusative, which typically expresses direction/movement (“into bed”).
in dem Bett (often contracted to im Bett) uses in + dative, which typically expresses location (“in bed”).
So:

• ins Bett gehen = to go to bed (movement)
• im Bett sein/liegen = to be/lie in bed (location)

• bevor is a conjunction that introduces a clause: Bevor ich gehe, … (“Before I go, …”)
• vorher is an adverb meaning “beforehand/previously”: Ich falte vorher das Handtuch. (“I fold the towel beforehand.”)

Because zusammenfalten is a separable verb/prefix combination in the present tense and simple main clauses. The conjugated part (falte) goes in the verb position, and the separable part (zusammen) goes to the end of the clause:

• Ich falte das Handtuch zusammen. In subordinate clauses or with certain constructions, it can appear together:
• …, weil ich das Handtuch zusammenfalte.

falten = “to fold” in general.
zusammenfalten emphasizes folding something up/into a compact form (“fold up”). In many everyday contexts, Germans naturally say zusammenfalten for things like towels, clothes, letters, etc., even though falten alone can also work.

It’s the direct object of the verb (zusammen)falten—the thing being folded.
In this specific sentence, das Handtuch looks the same in nominative and accusative, but you can tell by function: I fold what? the towel.
With a masculine noun you’d see the difference clearly:

• Ich falte den Schal zusammen. (den = accusative)

Yes, it’s required. German uses a comma to separate a subordinate clause from a main clause:

• Bevor ich ins Bett gehe, falte ich das Handtuch zusammen. Even if you reverse the order, you still need a comma:
• Ich falte das Handtuch zusammen, bevor ich ins Bett gehe.

Yes. Both are correct; the meaning stays basically the same, though the focus shifts slightly.

• Bevor …, falte ich … (sets the time context first)
• Ich falte …, bevor … (states the action first)

German often uses the present tense for scheduled or habitual future actions, especially with time phrases. Here it describes a routine: Before I go to bed, I fold the towel up. No future tense is needed.