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Please take a look at the article links provided below:

https://www.gardeningknowhow.com/ornamental/flowers/roses/roses-winter-care.htm

https://www.gardeningknowhow.com/ornamental/flowers/roses/mounding-roses.htm

I live in Colorado and do not ever cover my peonies here, I believe your climate is more mild so see no reason to cover them. For rosebushes I am provided a couple article links here for you: >> https://www.gardeningknowhow.com/ornamental/flowers/roses/mounding-roses.htm

https://www.gardeningknowhow.com/ornamental/flowers/roses/roses-winter-care.htm

You could also look up some nearby rose societies and Consulting Rosarians on the American Rose Society website and consult with them on best practices for your exact area.

If your tree rose is planted in a pot, it could be getting root bound and needs to be transplanted to a bigger pot. If this is the case be sure to check the root system when transplanting it as sometimes the roots will have started to grow together in a circular pattern around the inside of the pot. Try to pull them out of this habit a bit before placing it into the larger pot. If they are too tightly entwined, it is likely too late and the tree rose will eventually choke itself out. If planted in a garden or in the ground area, just keep it pruned up to stimulate growth and keep her well watered and fed with a good organic based rose foof. The Jobes rose food spikes seem to work well with tree roses too. Here is a link to an article I wrote about tree roses for you as well: https://www.gardeningknowhow.com/ornamental/flowers/roses/learn-more-about-tree-roses.htm

Leave it for now. Once it turns colder and the rosebush has gone into dormancy you can prune it back a bit. Pruning too soon will encourage it to grow more and that is not what we want going into winter.

I will provide you with links to some images of rose seedlings so you can check them against what you are seeing. At their very earliest stages they look alot like other first emerging seeds, so give them a little time to develop other leaves to be sure.

http://2.bp.blogspot.com/_tijE0VAnUsA/TEmcOqEksiI/AAAAAAAACNI/_uzQl_Q-mHU/s1600/RoseSeedlingBud2.JPG

http://farm8.staticflickr.com/7151/6508069513_f08c089b3b_z.jpg

Unfortunately your 12 year old rosebush may have partially died. When this happens, usually the top part of a grafted rosebush dies. The part we have enjoyed so much. The hardy root stock it was grafted to takes over and sends up it long canes which usually do not bloom for a couple years. When and if it does bloom the blooms are typically flat and not the color of the rose we desired and enjoyed. It is best to prune such roses down, dig them out and replace with a new rosebush that we like the looks of. Here are some links to articles for you to check out on this:

https://www.gardeningknowhow.com/ornamental/flowers/roses/own-root-roses-grafted-roses.htm

https://www.gardeningknowhow.com/ornamental/flowers/roses/no-blooms-on-roses.htm