One small example

Is this ROS (Rose of Sharon), if so..yes, it's a known invasive.

yep. It didn't thrive where we lived in California, but does just fine in WV. There are a few purple flowers mixed in the main shrub.
How do I control it?

Yes it’s ROS, Hibiscus syriacus.

Kill it on sight..that's the only control I know of.

Rose of Sharon can be a prolific self sower. Removing the unwanted seedlings is just basic garden maintenance. All weeds pull easier after a good rain as a start. There is the option of a chemical death of course. The right weeding tool can make things much easier as well and I would recommend having a look at the Cobrahead Weeder. Comes in short hand held and long hoe length handles.

Are the seeds a food source for birds?

If not, can I shear it after bloom to prevent seeds?

Since it blooms sequentially for a fairly long period, shearing wouldn't seem to be too profitable to me, but your mileage may vary. Deadheading would work, but be very tedious.

If you cut the trunk below ground level, even a little below, it won't regrow. If you leave one millimeter above ground, it will regrow.

That's good to know...because I get lots of resprouting.

One option is to consider growing a sterile varieties so you don't get seedlings.