Do you think driving with worn springs this will cause damage to the clutch/transmission?

The rear springs are very worn. Or maybe the drive train. It makes shifting difficult as the car squats if accelerated hard. And it feels as the hard is lurking back and for.

Nope

naturally worn suspension will result in a ride as youre suggesting

why don't you just replace the rear struts

its not hard

as for your trans - I always like to think about fluid first

after that you consider whether you need a new car

I rear struts i replaced, but the problem still is there but more bouncy, I think the springs are worn out. I been riding on them for a year now. Its gotten worst, Corning is very difficult also

Civics are front wheel drive.
Get your engine and gearbox mounts looked at.

No. But tires it will