Learn English – “at the time of”, it should be at the end or at the beginning of the sentence

Whatsoever is an adverb used for emphasis, and it means "at all."

I have no doubt whatsoever.

The sentence you quote means "she has no feelings at all."

As the NOAD says, whatever is also an archaic way to say whatever, but this is not the case in that sentence.

"About the children she abandoned" is a prepositional phrase describing the feelings she doesn't have. As in "I am thinking about you." about means concerning.

Both sentences are grammatical. Even in the link, Swan's grammar book says "usually go in the end position" (emphasis mine).

Of course, the most natural way is

Vasya did this homework at school. :)

The passive doesn't seem all that natural to describe or talk about such an everyday situation.

But since you asked, the first one

This homework was done by Vasya at school.

seems more generic and therefore natural than

This homework was done at school by Vasya.

I have not bolded at school as in your originals, because in either sentence one can stress either Vasya or at school, or both, at least when talking.

In writing, the term that comes first (Vasya or at school) seems to get more stress if read without deliberately stressing a word.