Learn English – A games console with the verb “play”

The form I have encountered is:- He plays games on his laptop. I don't particularly know why - it just seems idiomatic.

Of the three versions you have suggested, the first sounds wrong, as it suggests he is inside his computer. I would argue that the second and third are both correct, though the third has a subtle difference in meaning.

Where did you play basketball at?

This is okay. The at is not necessary, but it is normal in some people's dialects. You might want to avoid it, unless you want to sound dialectal. This advice applies to writing, not just speaking.

Where did you go to?

This is commonly used. For example: in the sense of trying to find a lost pet or even a lost object: Now where did you go to? Or Where did you get to? or Where did you go/run off to? It is a rearrangement of "To where did you do?", which sounds rather stuffy.

What did you do this for?

This is natural and common. 'What for' or 'for what' can have the same meaning as 'why', so this is fine and means the same as "Why did you do this?"

Which boy did you play games with?

is rearranged from the rather stuffy

With which boy did you play games?

As for

Where did you break in to?

Grammar Girl's explanation, in this case, is fine.

Where are you from?

is perfectly normal English. It would be weird and unusual to say or hear "From where are you?"

You want to know why Where are you from? is okay? This is one of those answers that amounts to because this is how we say it. Please note that in informal language it is acceptable to end a sentence in a preposition.