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First, make sure that it gets plenty of sun to reduce the flopping. You can also pinch plants back to keep them compact and cut them back by half in May.

The Chinese Sedum should do just fine.

Sounds like it might be a sedum. If you have a picture, send it back attached to this email and we can better identify it.

Yes, they should. Here is more information on growing this plant: https://www.gardeningknowhow.com/ornamental/flowers/stonecrop/stonecrop-plant.htm

You can, as it certainly wouldn't hurt the plant. Or you could wait until after the first frost and then cut the plant back to about a couple inches. Apply a fair amount of straw mulch to keep it well protected over winter. You can also choose to leave your sedum plants alone until spring (trimming back the same amount), allowing the foliage to provide the plant with winter protection.

This article should help:https://www.gardeningknowhow.com/ornamental/flowers/stonecrop/stonecrop-plant.htm