You can think of "flink" as being good/clever/skilled at something. It doesn't necessarily have the connotation of being smart or intelligent, because there are many things you can be good at without being particularly bright.
It can also be used in the "good girl" sense, as in "I think you're good because you acted the way I wanted you to".
Generally, I'd say that this is a tricky adjective to translate when there's no preposition involved. When the sentence is "Jentene dine er flinke [til/i] X" then it translates much better.
Not really. Flink can be doing what's expected, like your parents say you're "flink gutt/jente" if you do your homework/dishes without being told. You can also be "flink til å synge" which means a talented singer.
Street smart would be more along the lines of "kjapp i hodet", we don't really have a translation for it other than a direct translation of the actual word itself ("gatesmart"), and that's hardly ever used.
No, though it may have that implication in some cases. Someone who is "flink til å løpe" will of course be both fast and agile, but someone who is "flink i matte" doesn't need to be either of those things.
Speed will sometimes be part of what makes you "flink", but it's never a good translation for it.