Dear Mrs. H:

You wrote:

Is it necessary to include commas in the following text between ‘and/two, years/in’?
“She lives with her husband and two children aged five and two years in a suburb of N——”

No commas are needed. The conjunction and preposition do the job. If the phrase “in a suburb of N” began the sentence, then you would need a comma because it is separated from the word it modifies.

…and in the following text between ‘children/some’?
“is currently caring for three children some part-time.”

Yes, you do need a comma or dash here because the modifier follows the noun it modifies. This is somewhat unusual, but “some” clearly modifies “children” so that is what you go with. If the sentence were to read simply “part time” without the “some,” then it would be a simple adverb and there would be no need of a comma.

…and in the following text between times/such?
“They learn the importance of good hygiene practices and personal care at appropriate times such as washing their hands before meals and snacks.”

First of all, I would rewrite this sentence. If you are stuck with it because you are quoting someone, you should have a comma. The phrase beginning with “such as” modifies ”practices” and “care,” not “times.” Since the adjective phrase is separated from the words it modifies, it should be separated by a comma.

This run-on should be written as two sentences, something like the following:

They learn the importance of timing good hygienic practices and personal care. For example, they learn to wash their hands before meals and snacks.