"me" is not dropped after "with", no. Neither "me" nor any other similar personal pronoun is part of the idiom.
With "get", "over with" means "finished". This idiom is used in
circumstances where there is something undesirable or unpleasant about
the task to be finished, and the idea is usually that it will be a
relief to have it finished soon.
In brief
get it over with =
get it finished = be finished with it
-- We have to move all the furniture out on the lawn before the carpet layers come.
-- What a pain! Come on. Let's do it now. Let's get
it over with. Then we can have a snack while we wait for them.
CJ