Well, onions being a bulb and cold loving crop can be planted either in late fall or very eraly spring.
Separating clumps and planting can be done the same way.
I planted my onion sets, last december, here in Atlanta area(zone 8 as well), and got very good results. I will do it again.

Question:
I wonder if bouncing/walking onions produce seeds!?

alan8 are you still in touch with the person that gave them to you? Maybe they could give you information on how and when they divide them. I am in a different zone so I can't help you there.

There are different varieties of walking onions. I am now growing 2 different kinds and they seem to be quite different from each other. The 1st gets quite tough as it gets older, so I use the older ones as the "mother" plant. Little ones grow up around it and are tender and good eating. We eat the whole thing the same as a scallion you get at the store. They are good chopped up raw in anything you would like a little mild onion flavor. Salad, sandwiches, on potato, etc. They, when 2 years old, get flowers and will drop seed to grow even more plants the next year.

The 2nd type we grow have a very strong onion flavor so we use them more for cooking. They go into the dish at the end of cooking, so they don't get too mushy. We eat the whole plant of both kinds.

If your type is Egyptian walking onion there is some info in the link below that might be of use.

Hope this helps.

Here is a link that might be useful: http://www.egyptianwalkingonion.com/

Funny you should ask. I just planted some day before yesterday for the very first time. Here's some info from an old Organic Gardening "Secrets of Master Gardeners" booklet I have.
"Potato onions, (Allium cepa, Aggregatum Group), an heirloom variety, taste delicious and are great fun to grow. Their flavor is considerably milder than that of Egyptian onions. These multipler onions form large bulbs and small bulblets from each bulb. Plant them in fall in well-drained soil with plenty of organic matter, setting the bulbs 5 or 6 inches apart in the beds, with about half the bulb emerging from the soil.
Potato onions produce green bunching onions in spring and a crop of large (3 inches or mroe in diameter) bulbs that are excellent keepers in fall. Harvest the mature bulbs when the tops dry down. Just be sure to leave some bulbs and some bulblets in the ground for next year's crop. Full size bulbs left in the ground will each produce about 20 sets the next season, while the bulblets will mature into the season's harvest.
Hardy in zones 3 to 9.

This is all I "know", but by next year, I should "know" it all! (So to speak. :)

FYI: I bought my sets at our local Agricultural Coop. It's great. The owners actually pay somebody local to grow seed for all the traditionally grown vegetables for the area. Unbelievably cheap seed: 15 CENTS for a package of about 25 squash seed, for example. I've been counting my blessings all summer.

That may depend on what kind you have. Most all kinds of onions are very tough and can be divided and replanted most any time of year without killing them, but the exact best "how and when" to do that may vary according to type, as do when to harvest and how to use.

There is more than one type of onion called multiplier ~
Bunching onions ~ Allium fistulosum ~ Welsh onions, and those green onions you buy in buches at the store
Link One, Link Two, Link Three.

Potato onions ~ Allium cepa var. aggregatum ~ a shallot
Link One, Link Two, Link Three (look under "Features" near the end for a bit about potato onions, shallots, and walking/topsetting onions).

and, less often, Topsetting onions ~ Allium cepa var. proliferum ~ Egyptian walking onions, tree onions
Link One, Link Two.

From the above links, can you tell which are yours? Can you take a picture of the plants as they are now, then some of them dug up showing the underground structures? That would help us figure out which they are.

Question:
I wonder if bouncing/walking onions produce seeds!?

Nope, they produce bubils or mini onions that root in the soil as the tops supporting them fall over.

I started 3 years ago with 10 bubils and this year dug everything up and planted just 6. They produce a zillion bubils every year and will take over the planet if allowed to ;)

Still have a bunch of bubils in the shed for some reason. Most I put in the compost pile, but they won't likely compost, they will simply grow there.

Thank you for these links. My Ag Center doesn't know which I have, but I am going to try to identify them as time goes on. According to all the sources I read, these are the only kind of onions that really do well in the deep south.