Nice examples of a subordinate infinitive clause (proposition infinitive).

the object ("le bébé") itself looks like a subject, that takes a verb (as in "le bébé dormir") !

le bébé is indeed a subject ! It is that of the infinitive clause whose verb is dormir (which, like the name suggests, is in the infinitive mood). But then, the whole clause le bébé dormir is the object of regarde.

1 — yes, this is perfectly cromulent French. (edit : minus the question mark : Elle regarde le bébé parler le français.)

2 — Elle me regarde parler français. My guess is that me is used over je or moi because the subject of an infinitive clause was historically an accusative (in Latin), and me is the ‘object’ form of the first person pronoun – even though it is semantically a subject.

3 — If I understand your question right : since a clause can be an object, well yes – objects may very well comprise a verb !

On entend [les rames couper la vague]. (V. Hugo), infinitive clause

Je crois [qu’il est parti]. proposition complétive

There is also the infinitive group (groupe infinitif), which is distinct from an infinitive clause because it doesn’t have its own subject (and therefore is not a clause).

Je regrette de l’avoir dit.

I realize this answer could probably be made clearer, but hopefully these articles can help :

• https://capman.es/es/blog/infinitive-clause infinitive clause in English (for native Spanish speakers, but I think it makes the point clear!)
• https://www.francaisfacile.com/exercices/exercice-francais-2/exercice-francais-31153.php French course, looks very much like what French pupils learn in middle school

I hope this helps!