Learn Portuguese (Portugal): Ela fez uma careta e o bebé pôs-se a soluçar. - She made a face and the baby started sobbing.

While começou a simply means "started to", the expression pôr-se a adds the idea that the action started suddenly or abruptly. It perfectly captures the baby's immediate, involuntary reaction to the funny face.

Why is the pronoun hyphenated as pôs-se rather than coming before the verb (se pôs)?

In European Portuguese, object pronouns attach to the end of the verb by default in affirmative main clauses. Neither the conjunction e nor the subject o bebé acts as a "magnet" word, so they don't pull the pronoun se before the verb. If the clause were negative (e.g., using não), the pronoun would be pulled to the front.

Is fazer uma careta a fixed expression for "making a face"?

Yes. Uma careta translates directly to a grimace or a funny/weird facial expression. You will almost always see it paired with the verb fazer to express "pulling a face" or "making a face".

I usually translate "to cry" as chorar. What is the exact nuance of soluçar?