Thomas Hardy writes:
Oak was amused, perhaps a little astonished, and hanging up the hat in his hut, went again among his ewes.
Note 'Oak' is a surname here (e.g. Mr Oak or Gabriel Oak), and this quote comes from Far from the Madding Crowd.
We can highlight the relevant part of the sentence as
hanging up the hat in his hut, [he] went again among his ewes.
This sentence is parallel to the one you ask about. It is clear from common sense and the sentence construction that Oak first hung up his hat and then went out among his ewes (female sheep).
Hardy could have used having hung up but he didn't. And this should provide at least one example from real English to show your friend.
By sentence construction, we read English left to right and word order is important.
She finished her homework and went for a walk.
would naturally be taken, without anything else to hint to the contrary, to mean that she first finished her homework and then went for a walk.
An argument can be made that the word order (rather, phrase order) helps determine the meaning in the sentence you ask about, and in the one Hardy writes.
To show this let's reverse the order of phrases.
She went for a walk, finishing her homework.
It is natural to take the two actions as occurring at the same time, especially since both going for a walk and finishing her homework can take a long time.