Replacing Single switch with timer switch - Original two switches connected
It looks like that's a jumper wire that goes from the left to the right switch? The picture is a little hard to see the details. I would think it reasonable to guess that is the hot wire. When you go to swap out the switch, I would suggest replacing the jumper wire with something a little longer. I find trying to fight those shorter wires is just a nuisance and makes it more cumbersome and hard to tighten those wires down.

You should be able to duplicate the current setup with your existing switch. For a safer and better answer, maybe you can provide a picture of the timer wiring diagram and/or the back of the timer. Some timers are specific on where to connect the hot lead to, while others might not matter.

For your ground wire, yes, you can just ground it to the box directly, provided the incoming service is grounded to the box.

My final piece of advice... I remember being told the #1 cause for home electrical fires is the screws are not tight enough! Be sure to tighten them down!

I hope all that makes sense and didn't confuse you any further.