My husband also thinks these suckers are coming from the little green seedlings that fall off the tree then turn into cotton-is that true or are these suckers coming from the roots of the cottonwood trees? How do we get them out of our lawn?
Probably both! You can choose manual removal, or chemical means, but it will be a constant thing.
Do dead cottonwood leaves harm a lawn if left to decay? I have read that it is best to not rake up dead leaves, but I have been told cottonwood leaves have an oily substance that will kill grass.
This can be a way to suppress grass growth. Just make sure to rake it up if you wish to keep the grass in the area.
This article will help you to care for the tree:
The best thing you can do is cover them with a couple inches of hardwood mulch. Trying to plant among the roots will only damage the root system. Many of a tree's feeder roots are within the top 12 inches of the ground.
Here is more: