Szeretném, ha ma este korábban lefeküdnél.

Hungarian often expresses wants/requests about someone else’s action with a subordinate clause: Szeretném, ha + (conditional verb).
So instead of an infinitive construction, you get something like: I would like it, if you… (even though the real meaning is simply I’d like you to…).

Yes, ha literally means if, but in the set pattern Szeretném, ha… it commonly functions like that in English:

• Szeretném, ha korábban lefeküdnél. = I’d like (it) that you go to bed earlier.
  It’s not presenting a real condition; it’s introducing the desired situation.

Because after Szeretném, ha…, Hungarian typically uses the conditional mood in the subordinate clause to express a wish/request politely.

• lefeküdnél = you would go to bed (conditional), used here as a softened request.

Sure:

• le- = down (a verbal prefix)
• feküd- = lie (base of the verb)
• -nél = 2nd person singular conditional ending (you would)

So lefeküdnél is literally you would lie down → idiomatically you would go to bed.

Both exist, but Szeretném, ha… is the most common/idiomatic way to say I’d like (it) if…

• Szeretném is the definite conditional form: roughly I would like it (with an implied “it” = the whole situation/clause).
• Szeretnék is indefinite: I would like (more general).

In practice, with ha + clause, Hungarian strongly prefers Szeretném.

Hungarian often drops subject pronouns because the verb ending already shows the person/number.

• lefeküdnél already means you would go to bed, so te is unnecessary unless you want emphasis/contrast.

Yes, lefeküdnél is 2nd person singular (te), so it’s informal.
For formal you (Ön), Hungarian uses 3rd person verb forms:

• Szeretném, ha ma este korábban lefeküdne. (formal)

Because korábban (earlier) can be used absolutely/relatively: earlier than usual, earlier than expected, earlier than you’ve been going lately, etc.
You’d use mint only if you explicitly state the comparison:

• korábban, mint tegnap = earlier than yesterday

Yes. Hungarian word order is flexible and reflects emphasis (focus). Also, the verbal prefix (le-) can move depending on what’s emphasized. Common variants include:

• Szeretném, ha ma este korábban lefeküdnél. (neutral)
• Szeretném, ha korábban feküdnél le ma este. (slightly different emphasis; le shifts behind the verb)

Both are grammatical; the first is very natural here.