He is the caretaker.
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Basically (If I remember correctly) Hogwarts is a middle/high school and then you go straight from there to a sort of apprenticeship job then you get a job. No college.
He becomes an auror and the eventually the head auror. According to JK Rowling in an interview shortly after the release of the last book.
Not in the Harry Potter books one through seven. The last book ends with his final year at Hogwarts in which he can't become Headmaster because he has learning to do. If the author made another book about what happened after Harry finished Hogwarts, it is possible he could take over the job as Headmaster.
They probably thought his job wasn't good enough for a pureblood or something. Plus, the children hate Filch and the Death Eaters would have liked that. Plus, we don't know if they hated squibs or not.
Her future job was never mentioned.