can owners walk their dogs in this grassed R/W area?
I'm sure they can. But if you aren't sure, ask the association management or board.

But how do you prove which dogs are responsible
Unless there are witnesses, you don't.

which owners need to pay?
That's easy. The POA does the repairs and assesses all the owners if the common fund isn't sufficient.

That's how life works. Your taxes pay for services you don't use. Your insurance premiums pay for claims other people make.

It is not fair
Fair is where you go on rides and eat cotton candy. Has nothing to do with living in an association.