A couple of questions, what rose is this and is it own root or grafted? Also, how deep is that mulch around the rose? Is there much more rose underneath it? I brightened up the photo so it's easier to see and marked the three places to cut, I think, that third ones really hard to see. Start at the red line and snip it off at about a 45 degree angle about 1/4 inch above the bud eye. Look at the center of the cane. If it looks moist and a greenish white or creamy in the center it's OK. But I have to say that one, although nice and fat, looks iffy because the cane has died back awfully close to it. If it's tan or brown and rather dry looking go to the next line, yellow, and cut again. That one looks like it's good but if not then go to the green line. At that point you aren't going to have a lot of rose left if there's nothing below the mulch. Don't be fooled by that white tip. That's just dried and bleached from the sun. The wood itself is dead. Depending on whether or not it's own root or grafted the rose may not survive that deep a cut if you have to go to the green line. Or if it survives it may never be a strong healthy plant again. Own root roses have a better chance of survival and revival because they will send up new canes from the roots. Grafted roses not so much. We call them one cane wonders and they rarely come back to be strong bushy plants after dwindling this far. And if it is grafted and it appears to send up new canes from below ground those will most likely be the root stock sprouting and not your rose. I hope this helps and I really hope your rose survives!

Agree. Leave it alone. You are assuming that it will "become more bushy" if trimmed and that usually isn't the case since Early Girl is an indeterminate variety. If you want "bushy" plants you need to grow determinate varieties. Dave Here is a link that might be useful: FAQ: Determinate vs. Indeterminate tomatoes

The answer depends on individual preference. If you like the look of the flowers (or want to harvest seeds) then leave them alone until they are done blooming. If you think they are messy looking then cut them off. Many people suggest that cutting the flower scapes off will encourage the plant to redirect more energy into growing roots and leaves which will result in a more vigorous plant.

To start, we need to find out if it is really rooted... That should be quite simple. Does the 'rotted' part feel squishy? If so, that probably means it is rot. You will need to cut back all rotted parts to firm flesh.