Just remember that different plants seeds have germination times. Don't throw all away because some of them may still germinate. For seeds I normally use a very small plastic container or box with a lid to keep the moist inside. When the seeds germinate, after a while, I move them to proper pots. Otherwise, if you use normal pots for the seeds, they take a lot of space and you will be looking at an empty pot for a long time! :)

That first US map cannot possibly be for the Sunset zones!! There are 24 zones just for the western US and that only shows 12 in total. That looks more like just a variation of the USDA zonal map. Since so few references (or even gardeners) refer to the Sunset zones, I don't consider them of much value in this situation. In fact, even the USDA zones are not particularly helpful..........but location IS!! Just the general geographic area (i.e. southern California coastal, Florida panhandle, Long Island, Boston suburbs, Chicagoland, Philippines, UK, etc.). No need to get very specific.

Callery pears are notorious for producing root suckers. They can also self seed as well. Both are part of the reason they are considered such a nuisance tree. Removal of the tree would solve the problem. And many would encourage that course of action. If the tree remains, removal of the suckers as soon as they appear will be a regular part of your garden maintenance. Use pruners rather than trying to pull and cut them off as close to their source as possible.....sometimes this will require being below the soil surface. There is also a product you can buy that is intended just for this purpose. It is called Sucker Stopper and is a synthetic plant growth hormone. It is used extensively in commercial orchards but I have no personal experience with it and have no idea how well it works.

Thanks for the advice! The mulberry shrub is already disposed of, and yes, I did notice the yellow roots! We were planning to move this year ( which got delayed ) hence the plant in the pot, for the new place. Yes it self seeds here as the soil under this shrub allows it. However at my mom's place her rose of sharon was on the front lawn surrounded by grass, never an issue with self seeding and spreading. That's the any way to go in future.