Ah, I see you are using manually scheduled tasks. Firstly, yes setting a deadline does set a date -- but the Move Project command will also you to move the entire project and maintain the relative deadline.

Be aware that "setting a date" through automatically scheduled tasks still allows you to move tasks -- the issue is setting date constraints by typing in dates. If the duration of task 9 increases, the start of task 12 (successor) will change automatically.

Task number 10 right now is aligning with the finish of the summary task -- but the summary tasks duration has been manually entered. Why should it take 60 days if the only true subtask is 3 days in duration?

Are you trying to say that task 9 only is 3 days of work but you want to allow 60 days to complete it?