"Will the stem divide at the cut and produce a double harvest of flowers that same season?" That's correct - if you cut back early enough several shorter stems will grow from the leaf node below where you cut and you will have more flowers. In zone 5, at this time, it may be too late for this to work as buds are already setting and you risk losing your bloom altogether.

I've had Sentimental Blue for many, many years and have many, many established clumps of them (makes a beautiful, tidy edger). A few additional another flower heads will pop up here and there post dead-heading, but they are not really considered re-bloomers (at least not reliably). Once the main flush is done, that's pretty much it for the most part, I view any additional flowering as a bonus :0)

This was from a packet of seeds sown about 10 years ago. They were supposed to be white and only 10-12 inches tall. Unfortunately I don't remember the variety, though I guess it doesn't matter as they were obviously not what the packet said. I did get some lovely whites but all were at least 2ft. Here is a link that might be useful: balloon flower

My sedums aren't blooming yet, but they were taken from plants just like you said. I stuck them in the dirt and kept them watered. I cut back bee balm, shastas, even my hibiscus. It makes for a fuller plant and lots of blooms. Almost all perennials can tolerate this. Great too for staggering bloom cycles to extend the flowering time. Cut 1/3 of the plant when you see buds. Cut another 1/3 a couple weeks later or just as the buds are opening. Leave 1/3 to bloom as normal. It also staggers the height and helps hold up the tallest stalks that weren't cut back. The Well Tended Perennial Garden. Can't go wrong with that book.