Learn English – Is “now as well” correct

Your second set of sentences is more colloquial, but the only real error in the first set is the placement of "always". The ordinary locations for always are

• after BE or HAVE or the first auxiliary in a verb construction (as in your example), or
• before a single verb other than BE or HAVE.

It is placed at the end of the sentence only as an afterthought, or for extra emphasis, so when it is placed there it must be set off with a comma or dash, reflecting the vocal pause before it is added:

It is capitalized—always.

The other differences between the two sets are not errors, but bear different emphases.

MYSELF

• Xself at the end of a sentence is roughly equivalent to too; it may express a little more emphasis, or stress personal participation.

  The King is looking into this himself, not having a servant look into it.

• Xself set immediately after its referent, X Xself, is emphatic: it is roughly the equivalent of even X. It is employed when the referent's involvement is surprising or impressive:

  • The King himself is looking into this, though you might have thought he wasn't interested.
  • The King himself is looking into this, which tells you how important we consider the matter.

When CLAUSE
The placement of when used to express myself depends on the context.

• If the most important thing in the sentence is the capitalization, is always capitalized should come immediately after the subject, as in your second version. In that case, when used to express myself becomes an important but secondary qualification of the general statement

• But if the most important thing is distinguishing when 'I' is capitalized, that should come before the predicate, as in your first example, and it should be set off with commas or dashes:

  'I' in English, when it expresses myself, is always capitalized.

  In this position it 'interrupts' the ordinary Subject-Predicate order, which gives it extra prominence.

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.