If I ask a question in Past Simple using the coordinating conjunction and, does the second verb take -ed or not?
e.g. Did you muster up the courage and propose? or Did you muster up the courage and proposed?
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You don't need the verb -ed' as you say. Why? Because when you have Did... it covers it!
Did you muster up the courage and propose (to her/him)?*
The and joins the two actions and thus, they go in the same flow. Also, you are asking the person for the past action of having courage and proposing someone as well. So, no '-ed.'