Flowers - you can remove them at any time for paniculatas like Limelight. However, some people leave them "on" or winter interest. Others cut them when they change from white. Others cut them off when they turn brown. So it is up to you to choose. Nothing is "required". To deadhead flowers, you can cut the little string that connects the bloom to the stem. So cut the string, not the stem.

No need at this point to wrap in burlap as Limelight is very winter hardy. You can wait until mid-to-late May to prune all the way down those stems that did not leaf out. Some people do prune it down to -for example- 1' to 18" since it has a habit of growing back a lot from one year to the next.

Continue monitoring bent branches to see if they re-orient. If they do not look aesthetically pleasing after a while and the stems are not inclined to self-correct then consider pruning them. Winter would be a good time for this since all the foliage has dried out and you clearly see the stems without the leaves getting in the way.

Pluto, you are correct that pruning half the height refers to shrub form panicled hydrangeas like Limelight, not the tree form that you have. Since it's a new plant this year, other than removing the spent blossoms when you want to (either now or next spring), I wouldn't do much pruning of the top of the plant. (Your friend can use floral wire to provide "stems" for the craft project.) I usually give plants a minimum of a year in the ground before pruning since the more leaves it has to create food, the quicker it grows roots and settles in well. If you have sprouts along the "trunk", prune them off so that it stays in tree form, and in future years if trunk sprouts start, you can just rub them off gently as soon as they appear. If you have branches that are crossed and rubbing or badly shaped, you can prune them, but use a light touch for now. Do any pruning to just above a bud since new growth will start from the bud and you don't want a dead stub.

Thanks for the answers! Is there anything else I should do to prepare it for winter? Fertilizer? Extra mulch on the base? It's in a brand new flower bed that currently has about 2" of shredded wood mulch.

I don't typically use fertilizer unless I know my soil is lacking something. 2" of mulch sounds good.