You can support your peony with a tomato cage. Reduce the height of the cage as needed cutting off the bottom section. Use a wire cutter to cut off the section. To your second question, Peonies only bloom on new growth that will emerge in spring. Except tree peonies, the visible top growth will die out in the winter. I usually prune off the scraggly looking leaves in fall when I do the fall clean-up. By the way if the peony eye that are forming underground is not subjected to the winter chill, they will not flower.

There are also Gro-thru grids intended specifically for this purpose

I use grow through round grates, but be sure to install them as soon as you see the new growth in the spring, likely early May. Also get sturdy ones which are high enough, as different varieties grow to different heights. Try Lee Valley, great store for all your gardening needs and you can shop online. Canadian company but will ship to US.

For most rather than early May which may be a autocorrect use early March. You need to place any supports so that the stems grow through them. A suggestion depending on the cultivar is to remove the outside flowers while they are soft to the touch but not open. Although your plant may not be large enough the outside stems without the flowers can provide support for the internal flowers to at least not touch the ground.