Learn Swedish: Jag måste springa, oavsett hur trött jag är.

Here, oavsett (regardless of) pairs with the interrogative word hur (how) to mean "no matter how". This combination introduces an indirect question that acts as a subordinate clause expressing concession, meaning the action in the main clause happens regardless of the condition.

Why is the word order jag är instead of the usual question order är jag?

Because hur trött jag är is an indirect question acting as a subordinate clause. In Swedish subordinate clauses, the subject (jag) always comes before the verb (är). If it were a direct question standing alone, the verb would come before the subject: Hur trött är jag?

Does the adjective trött still need to agree with the subject in this type of structure?

Yes, adjective agreement applies exactly as it does in a standard sentence. If the subject were plural, the adjective would take the plural -a ending: Jag måste springa, oavsett hur trötta vi är (no matter how tired we are).

Could I use trots att instead of oavsett here?