Learn German: Wen ich abweise, der bleibt draußen, obwohl er reich ist. - Whomever I turn away stays outside, although he is rich.

The case depends on the person's role in each specific clause. In the first clause, the person is the direct object being turned away (abweisen), so we need the accusative Wen (whomever). In the second clause, that same person is the subject doing the staying (bleibt), so we use the nominative demonstrative pronoun der.

What is the word der doing in the middle of the sentence? Is it just "the"?

Here, der is a demonstrative pronoun meaning "that person" or "he". When a sentence starts with a free relative clause like Wen ich abweise, German speakers often use der (or dem, den, etc.) right after the comma to smoothly connect the "whoever" or "whomever" person back into the main sentence.

Why isn't the separable verb abweisen split into weise ... ab?

Because Wen ich abweise is a relative clause. Relative clauses act like subordinate clauses, which means the conjugated verb gets pushed to the very end. When a separable verb is sent to the end of a clause like this, its two parts remain attached as a single word.

Why is the verb ist at the very end of the final clause?