Learn English – “the hotel where you stayed” or “the hotel where you stayed at”

Where and when, either as interrogatives or as relatives, are not pronouns but pro-adverbs—they ‘stand for’ adverbials of place and time, respectively.

The place where I found it = the place at/in which I found it
Where did you find it? = At/in what place did you find it?
The time when I found it = the time at which I found it
When did you find it? = At what time did you find it?

It is thus not strictly true to say that “a relative clause always contains a relative pronoun”—it may also be headed by a relative pro-adverb. And in some cases (as in He told me to stop, which I did) the relative ‘pronoun’ is actually a component of a relative pro-verbal construction with DO.

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.