Vos bijoux, puis votre portefeuille. Ne criez pas!

All are possible, depending on context.

You can't change the owner in the middle of the sentence without notice, since the possessives are identical for "il" and "elle":

• her hat, then his coat = son chapeau à elle, puis son manteau à lui.

The difference is that in English, "coat" is frequently used as a generic term, but "manteau" is not a generic term. I.e., a "manteau" is never a jacket. "Un manteau" is specifically a long coat, extending anywhere from mid-thigh to the ankle; it may be thought of as an "overcoat". "Une veste" is a jacket (also called generically a "coat"), a garment that extends either to the hip or slightly below.

a jacket (une veste, un blouson) is short, a coat is long.

These moderators are so extraordinarily pedantic. Jacket and coat are used interchangeably in English. A long "manteau" would absolutely be referred to as a jacket.

That would not change the meaning but it would be less usual.

Another usual adverb is "ensuite" in this case.

While the order is correct, before is "avant." The exercises are to teach you the words and how they can be used. Once you go into "avant" vs. "puis" it really comes down to context, which this exercise doesn't provide.

The point is that "his hat, and then his coat" would back translate to "son chapeau, et puis son manteau".

"puis" is a matter of time, and means "then".

"next to" is a matter of location in space and it translates to "à côté de, près de".

You would have to adjust the French sentence if you wanted to change the owner in the middle:

• son chapeau à elle, puis son manteau à lui.

Otherwise, the two objects are conventionally supposed to belong to the same person.

It is accepted as correct now if it wasn’t then.

"His hat then his coat" and even "its hat and then its coat" are also accepted, as long as you don't mix the pronouns in a single sentence.