What you did was all wrong. Replant any clean, firm sections there may be immediately in suitable sunny, open place. If all is spoiled or diseased looking discard the whole mess.

You want newer parts with strong-looking shoots and roots. Break off older, inner parts of the clump. New sections only a few inches long may be adequate as long as they are of good quality. When you purchase named bearded irises the sections provided may be quite short.

Search internet for "bearded iris propagation" and the like for more details and illustrations.

Assuming that all of the rhizomes are still in good shape, is it possible to store them until spring? Does anyone have experience with storing the rhizomes?

You can buy them for planting in spring as well as fall, but the basic life cycle is new roots in fall, new shoots in spring with flowering in spring and then a dormant period in summer. The wild ancestors are from semiarid climates.

One bought packaged are quite dry. If you have to keep them over go through and select good sections, store these cool and dry. Monitor for deterioration and infestation during the winter. As with digging and replanting a web search for something like "bearded iris storage" seems likely to generate at least some hits.

Bearded iris can't be treated like other 'bulbs'. They are dormant in summer, not winter. Dahlia tubers can be stored over winter, but iris rhizomes cannot.

Plant those suckers now. They're pretty tough, so you probably didn't do any real harm, but they won't survive the winter in damp peat in a greenhouse. They will rot.

If you dont' have a garden space for them you could plant them in pots. They like to be planted very shallowly, like with the top surface of the rhizome exposed. First time I planted bearded iris I planted them below ground like you'd think they wanted. They pushed themselves back up to the surface.

It seems likely to me the rhizomes supplied to garden centers in spring by wholesale sources were dug in fall along with those dug for fall sales.

Bearded iris are remarkably durable plants, probably second only to daylilies in the amount of neglect/abuse they will tolerate :-) That's why you often see them still going strong in vacant lots and abandoned homesteads. I've seen them thriving and blooming on dump piles, where they've just been discarded and tossed by a previous owner.

It is certainly not too late to plant them in the ground - that would be the preferred solution. If necessary, they can also be potted up and overwintered in containers, either out in the open or an unheated greenhouse. Potting soil is preferrable to peat, but very slightly moistened peat would work also. I agree with bboy - the toes you find for sale bare root in spring are from fall digs and have been stored through winter exactly this way.

I wouldn't expect the growers to be going out and ripping up a second set for spring sales, and the ones I see offered look like they have been around since fall.

As gg said, they're definitely tough. This past summer I dug a clump from DS's property, stuck them in a garbage bag to bring home, forgot the bag when I left and came across it again 2 weeks later. Work was being done in the front yard and I guess no one ventured to the back!

I found only a fraction had rotted from the heat and moisture (bag was under a sprinkler system) so I brought them home, cleaned, clipped and sprinkled them with fungal powder and put them in a bucket on my covered deck.
Again.......they were forgotten for several week (new granddaughter!) but they were nice and dry and some started sending up shoots. Got them in the ground in Sept. and there are still a few green sprigs. Don't really expect blooms next summer, but we'll see!

Wild ancestors are semiarid climate plants that spend long hot summers sitting on top of the ground baking. Tulips come from the same conditions and have therefore the same seasonal cycle. Of course, these end up pretty deep in the ground but can also sit around out in the air for long periods during the summer dormant season.

Of course, realize that the newer types of bearded iris run circles around the older models. With the exception of Great Lakes, which is intensely fragrant, are these Iris really worth the trouble?
I'd suggest contacting someone who is a hobby Iris grower, "maybe on the forum" and starting with some really good ones. Some of the older types are pretty dull.
Ditto on getting them out of that bucket. If you want them to live, they'd survive better outdoors in a pile than in the GH.

Re-blooming cultivars much better value. Not only do they flower twice per year but it seems there may be a tendency for the foliage to hold up better during summer.