Стоит мне сделать скриншот, как сестра просит показать его ей тоже.

No. Here как does not mean how.

The pattern стоит + dative + infinitive, как... is a fixed construction meaning:

• as soon as...
• the moment...
• no sooner... than...

So Стоит мне сделать скриншот, как... means something like As soon as I take a screenshot, ...

Because this construction uses the dative case for the person who performs the action in the infinitive.

So Russian says:

• стоит мне сделать
• стоит ему сказать
• стоит нам выйти

This is just how the pattern works. It is best learned as a whole chunk: стоит кому-то что-то сделать, как...

Not really. Although стоить can mean to cost or to be worth, in this sentence стоит is part of an idiomatic construction.

So you should not read Стоит мне сделать скриншот as It is worth it for me to take a screenshot.

That would be a different structure:

• Мне стоит сделать скриншот = I should take a screenshot or It would be worth taking a screenshot

But:

• Стоит мне сделать скриншот, как... = As soon as I take a screenshot, ...

Because сделать is perfective, and here Russian wants to present the action as a single completed event.

The idea is:

• I do one complete action, take a screenshot
• immediately after that, something else happens

That is exactly the kind of situation where perfective is natural.

If you used делать, it would suggest more of a process or ongoing activity, which does not fit as well here.

That is simply the normal Russian collocation.

Russian commonly says:

• сделать скриншот

English uses take, but Russian prefers make/do here. This is very common across languages: the noun is the same idea, but the accompanying verb is different.

Because the sentence describes a habitual or typical situation.

The meaning is not about one single event. It means something like:

• Whenever I take a screenshot, my sister asks to see it too

Russian often uses the present tense for things that regularly happen.

So просит here means:

• asks, in the sense of keeps asking whenever this situation comes up

The person being asked is understood from context: me.

Russian often leaves out an object if it is obvious. So here:

• сестра просит показать = my sister asks me to show

A fuller version would be:

• сестра просит меня показать его ей тоже

That version makes the missing меня explicit.

Because скриншот is a masculine noun.

So when you refer back to it with a pronoun, Russian uses его for it.

Here его means it, not him.

This is normal because Russian pronouns reflect grammatical gender:

• masculine noun → его
• feminine noun → её
• neuter noun → его

Ей is the dative form of она, and after показать it means to her.

Тоже means too or also.

So:

• показать его ей тоже = show it to her too

The idea is that the sister wants to be included as well.

Yes, Russian word order is flexible, and moving тоже changes the emphasis.

In this sentence:

• показать его ей тоже emphasizes to her too

But compare:

• сестра тоже просит показать его ей = the sister also asks
• сестра просит тоже показать его ей = emphasis on also asks to show

So the position of тоже tells you what exactly is being marked as also/too.

Yes, it is understandable and basically natural.

A few speakers might prefer to make the omitted object explicit:

• Стоит мне сделать скриншот, как сестра просит меня показать его ей тоже.

Another slightly neater option is:

• Стоит мне сделать скриншот, как сестра просит показать его и ей.

But the original sentence is perfectly understandable and uses a very common Russian pattern.