Welcome to the forum, newgardener.
I'm fairly new to rose growing too, but, have learned a lot from this forum ( great folks in here) and from videos and reading.
Pruning at this time of year depends on where you are located. I'm in zone 6 so we get cold winters. I would not prune in fall or late summer because the new growth that roses put out after pruning would likely die in winter. If you live in an area that doesn't get cold winters I would think you could prune some of the diseased and old wood off now as some of the stems you showed don't look healthy.
I'd advise you to find some videos that show how to prune for your garden zone before trying it. The best time to move an established rose plant is when it's dormant. They go dormant in the winter here, so, again, it depends on where you live.
Let's see what more experienced people have to say about your situation before you listen to me.
