Learn Polish: Prawdę mówiąc, wolę zostać w domu. - To tell the truth, I prefer to stay at home.

It literally translates to "truth speaking." Mówiąc is a present adverbial participle (meaning "speaking" or "telling"), and prawdę is the accusative form of prawda (truth) because it acts as the direct object. Together they create a set framing phrase.

Is prawdę mówiąc a fixed phrase, or could I change the word order to mówiąc prawdę?

Prawdę mówiąc is a fixed idiom used to frame your attitude, much like "to tell the truth" or "frankly" in English. If you change the order to mówiąc prawdę, it loses this framing function and sounds like a literal description of two concurrent actions, meaning "while speaking the truth, I prefer to stay at home."

Could I use szczerze mówiąc in this sentence instead?

Yes, absolutely. Szczerze mówiąc (frankly speaking / honestly) is almost completely interchangeable with prawdę mówiąc here. Both are participles used to frame the main clause with the speaker's honest attitude.

Why is the perfective verb zostać used here after wolę instead of the imperfective zostawać?