Oldhouselover: Concerning the tree, cut it whenever you would like or find convenient. It's already growing there, it won't hurt to leave it until fall when the leaves on the roses are cleared and it's easier to reach in there. Don't sweat it or worry :). Just make sure not to forget to cut it in the fall - it must go. 4 more months of growth from the tree won't hurt the rose. Amazingly, if that is a Tree of Heaven, which I'm almost positive, that tree may be only one year old, possibly two. They grow fast. So don't forget to cut it out this fall :). I personally would not use any herbicide whatsoever on the tree of heaven. Due to your sentimental feelings for the rose and desire to keep it as healthy as possible, the risk of accidentally spilling even a little herbicide is too great. Just be vigilant and be ready to cut the little shoots the ToH will inevitably send out in the spring once you cut it down. I would move the hostas. Others here would have better knowledge on how/when to trim the rose to a more managable and visually appealing form. The hostas, though, are being overrun by the rose and if you would like to save them, move them. Even if you want to keep them in that bed, you can move them out of the way of the rose. It happens. Sometiems my stuff gets out of control so I just move it. It's hard to kill hostas. They'll be happy wherever you put them as long as it's not full sun. Most importantly, don't OVERCARE for the rose!!!!!! :). That's the biggest mistake I see most people on all of these forums make, especially people new to gardening. I've seen more isntances of people overcaring for their plant and damaging it than undercaring for a plant. Keep in mind, the rose is decades old and apparently no one cared for it too much, and it's still alive and thriving :). Ignore your impulses for fertilizer, watering everyday, and all that. It is a beautiful bush and I'm sure those on this forum will be able to help you trim it into a more managable form :).

You haven't gotten a knowledgeable answer on specific rose varieties. So here's some info on what to look for in various catalog descriptions. For tea and similar uses, you want to look for varieties described as having large hips. For rose water or syrup, you want to look for varieties that are listed as intensely fragrant. They probably will not be the same varieties. But there should be climbers that fit both categories. Best I can do....

Perhaps Graham Thomas, hybridized by David Austin. Many "Austins" (as roses hybridized by Mr. Austin are called) shatter, or lose petals, quickly after opening. If you want to cut them to bring inside, cut while they buds are still tight, you might get it to last a day or two. Here is a link that might be useful: Graham Thomas info at Helpmefind.com

The problem is that if it is a once blooming rose it will only bloom on OLD wood or LAST YEARS growth. If you keep cutting off the new growth off it will never bloom. Without seeing a bloom it would be impossible to ID it for you. Is there anyway you can put a trellis behind it and let it grow on that so it doesn't touch the house?