Others have recommended using a coffee filter and wood or white glue. That should work but if the two edges fit nicely together, you can apply some thinned white Elmer's glue to the edges with a toothpick and gently work them together until the glue gets tacky.
The cones appear to have water stains so they are not particularly valuable anyway so don't overthink it.
Unless you are a woodworker or extremely lucky, anything you do there will not improve the appearance much. I would just apply a light stain to the corner so that it doesn't jump out at you.
Leave the boards alone. The caps are more than likely fine. "Recapping" is the go-to "upgrade" for those who just want to do something. If you want to know if it's worth the effort, get a microphone, software, etc. and measure the speakers before and after the "upgrade". Otherwise, you will never know if it was effective or not.
That's a good plan.
As they are now, about $75 to $100. If you do the suggested repairs, about $75 to $100
