Learn Swedish: Mannen hänger jackan bakom dörren. - The man hangs the jacket behind the door.

No, it stays exactly the same. You might remember that some Swedish location words change depending on whether there is motion (like in vs inne). However, extended spatial prepositions like bakom (behind), framför (in front of), and under (under) do not change forms based on movement.

Does Swedish use the same verb for a person hanging a coat and the coat itself hanging on a hook?

Yes, at least in the present tense! You can say mannen hänger jackan (the man is hanging the jacket) and jackan hänger bakom dörren (the jacket is hanging behind the door). The verb hänger works for both the action and the state of hanging.

I sometimes mix up the endings for 'the'. Why is it jackan but dörren?

Both are en-words, but they follow slightly different spelling patterns. Dörr ends in a consonant, so it gets the standard -en ending to become dörren (the door). Jacka ends in a vowel. For en-words ending in -a, you drop that final -a and add -an to make it jackan (the jacket).

If I wanted to emphasize the location and start the sentence with bakom dörren, what happens to the word order?