When I check off a completed task, will it disappear?

Nope. I, for one, like to see a nice list of completed work. Not to assume you are compulsive like me, but… it does feel nice, doesn’t it?

So, if it's checked "Done" it just shows up with a line through it, and it isn't included in any re-scheduling.

If it's checked off as "Remove" then it will be completely removed from the planner.