Photos are right side up for me :-)

What you are showing is the flowerhead of a mophead type of bigleaf hydrangea, H. macrophylla. Almost impossible to determine which one, as soil conditions and amount of sunlight will affect flower color, sometimes radically.

Many gardeners leave the aged flowers in place as they continue to have an attraction even well beyond peak bloom. And removing them will not encourage new flowering. But if you do want them gone, you can just deadhead them, clipping them off just above the next set of leaf axils. Don't prune back further as that will impact next year's flowering (buds are aready being set). Any dead or leafless stems can be pruned back whenever you like, but usually as the shrub is just leafing out in spring. In our climate it will be easy to tell as the leaf buds start showing and swelling by March at least.

Thank you! I‘m trying to get a larger photo to post as the hydrangea looks “skeletal“ with lots of wood showing (including some dead branches I’m sure) and I’m wondering how to handle. I’d like it to look full next year. Let me see if I can get a another photo to attach:

Thank you! When (what month) is the best time to move? and should I be looking for a site with full sun or partial or?

Anytime from mid-late fall to very early spring to move. And ideal siting is where it would receive afternoon shade, at least from 2pm on. Morning sun is fine.

Thank you!