Isn't it frustrating! There are lots of varieties out there now that are advertised as "landscape" peonies. These are the ones with really strong stems that don't flop like the ones you're talking about. The only way to keep the one you have now upright is to use a peony ring for support, best put out in the spring when it shows above ground so the peony can grow up through it. Some peonies that are in place for several years are too big for peony rings, though. I've heard of people making cylinders from lightweight metal fencing to place around their peonies to do the same job on bigger plants. Also you can use a tomato cage cut in half, top half for smaller, shorter plants, bottom half for larger ones.

As for the tree peony, as far as I know there's no way to correct a downward facing bloom, sorry. You could cut a few to float in a bowl inside--I know that's a crappy substitute when you want to walk in your garden and enjoy the blooms, though.

One thing you might want to try is Azomite, it's supposed to make the stems stronger. I used it last fall and this spring, can't say that I noticed a difference. If you try it and it helps post back! I also tried fertilizing my peonies with BulbTone this spring, an organic bulb fertilizer suggested by Kousa. The blooms were really big but it didn't do anything to improve stem strength, think it made the situation worse. I've never had bigger, prettier blooms, though!

Another thing you can try if you didn't get the peony rings in place in time is the individual bloom supports, just a thin wire stick with a loopy bit on the end. Shove the stake in the ground, hopefully avoiding impaling the roots, and place the stem close to the bloom in the loop. They do work pretty well to support individual blooms and aren't incredibly obvious since they're green in color. They're generally available at Walmart or any of the big box stores that have a gardening section, same area as the plant rings.

BulbTone does not strengthen stems. I've used it on several plants including mixing it or bone meal in with my dahlias when I plant them for several years now.

Another option, which I use with my gladiolus is those green coated metal support poles. I tie brown twine or green gardening ties (blanking on the name) that is like a giant bread twistie essentially that comes on a roll several feet long from pole to pole around the planting area.