I think I can figure out how to prune it back, and let it recover next
spring.

Forsythia is pretty durable. So you can prune it back pretty hard,
and it will recover.

What I don't know is how to get rid of the secondary roots. It has
arched over and rooted itself in areas that I want to return to lawn.
Do I have to dig up the roots? How deep do I have to dig?

Never done this with a forsythia, but you can probably just pull them
up (if they come out), or dig enough to get out the main clump of
roots. If they keep coming back, keep pulling them up. Well, you'll
have to do that anyway, if it is like most forsythias (at least by
reputation, ours is in a shady spot and was pretty overgrown by other
plants including poison ivy, so we haven't yet experienced this
firsthand).