Yes, they do spread their footprints by suckering. It's a slowish process, more similar to lilac than to sumac. Keep the seeds moist packed right up to time of planting...their viability declines rapidly if allowed to dry out.

I learned that the hard way the 1st year I gathered them. I put them in the garage & they all shriveled on me. not at all like the buckeyes on trees. now I plant them right away or store them in a mix of damp sawdust & potting soil. the harvest is nearly done for this year. I do not pick them until the husk starts to open. the trick is getting them before the squirrels do. I shot a squirrels yesterday with one in it's mouth.