Learn Hardcore German: Manchmal schlafe ich auf dem Sofa ein.

Because einschlafen is a separable-prefix verb in German. In main clauses the prefix (ein) detaches and goes to the end, while the finite part (schlafe) stays in second position.

What’s the difference between einschlafen and schlafen?

Einschlafen means “to fall asleep” (the moment you drift off), whereas schlafen simply means “to sleep” (the state of being asleep).

Why isn’t there a reflexive pronoun, like ich schlafe mich ein?

Einschlafen is an intransitive, non-reflexive verb. You don’t need (and you can’t use) a reflexive pronoun here—you just say ich schlafe ein.

Why is auf dem Sofa in the dative case?

The preposition auf can govern accusative for movement or dative for location. Since you’re indicating where you fall asleep (static location), you use the dative form dem Sofa.

Could I say auf das Sofa (or aufs Sofa) instead of auf dem Sofa?

No. auf das (contracted to aufs) implies movement “onto the sofa.” Here you’re not moving to it but simply falling asleep where you already are, so it must be dative dem.

Why is Manchmal at the very beginning of the sentence?

German main clauses follow the verb-second (V2) rule. If you put an adverbial like Manchmal first, the finite verb (schlafe) still occupies slot two, and the subject follows it.

How would the word order change if the subject came first?

If you start with the subject, you get:
Ich schlafe manchmal auf dem Sofa ein.
The finite verb remains second, and the adverb manchmal moves after the verb.

Can I drop the article and say auf Sofa instead of auf dem Sofa?

No. Sofa is a countable noun here, so you must use the dative article dem. German doesn’t allow dropping the article in this context.

How would you express “I fell asleep on the sofa” in the perfect tense?

Use sein as the auxiliary and the past participle eingeschlafen:
Ich bin auf dem Sofa eingeschlafen.
Or with manchmal: Manchmal bin ich auf dem Sofa eingeschlafen.