two simultaneous events with the same subject?

Question

Source 1 : Compact Advanced Student's Book

There is an explanation in the book.

two simultaneous events with the same subject; the participle clause normally comes second

Example:

The woman hurried after her dog, calling to him to come back.

Source 2 : Objective English For Competitive Examinations

When two actions occur at the same time and are done by the same person or thing, we can use a present participle. We can express the first action with a present participle.

Example:

Singing to himself, he walked down the street.

He sang to himself as he walked down the street.

In the first expression, it said "normally comes second".

In the second expression, it said "the first action". Why?

Based on these expressions, can we write this sentence like this?

The woman hurried after her dog, calling to him to come back.

While hurrying after her dog, the woman called to him to come back.

or

Singing to himself, he walked down the street.

He sang to himself , walking down the street.

Answer 1

Syntactically speaking, it's irrelevant whether the (adverbial) "participial phrase" is one word or several, so let's look at a really simple example...

1: She ran away, sobbing.
...where the adverbial can be "fronted" to give...
1a: Sobbing, she ran away.

Note that #1 is the default sequence in English (as OP's textbook says, the participle clause normally comes second). Probably it's precisely because this is the default that the comma separating the two components of the utterance is effectively optional in this version.

But of course we can reverse syntactic structures themselves, with...

2: She sobbed, running away.
2a: Running away, she sobbed

It may or may not be obvious to non-native speakers that those two are a little "unusual", so it's worth pointing out that there's nothing remotely odd about any of...

3: She smiled, sipping her drink (discarding the comma would be very odd here)
3a: Sipping her drink, she smiled
4: She sipped her drink, smiling
4a: Smiling, she sipped her drink

In most cases of this general type, one of the two "simultaneous activities" is more obviously the primary one (running away and sipping a drink) that we can adverbially modify with the secondary activity expressed as a participle (sobbing, smiling).

It's just that syntactically, we can reverse the constructions if we want - which may sometimes change the nuance. For example, a "fronted adverbial" usually draws more attention to what would otherwise have been a "secondary" element.

Answer 2

All those versions are correct. Both books are technically correct as well.

The first is correct because we do normally put the participle clause second. It's not a rule, but it's the default form. The second book is also technically correct but misleading because we can express the first action with a present participle, but this phrasing incorrectly implies that this is the normal way to do things.