Question 1. I am building myself a lamp and I am using a plastic 1 gang outlet box. There are no ground wires. How can I properly ground the lamp?
Question 2: Can I attach a metal bar and place it on the wood, then attach the ground wire?
You get a usb charger, and wire up the lamp in 5V, going to LEDs. (There are 5V LED bulbs, or you could hardwire them in, depending on the type of lamp you're building.) There is no ground wire, but it's low voltage, so there's no safety issue.
EDIT: 12V is also good; you can use whichever is most convenient.
At least in the US, it is normal for lamps to not have ground wires. As long as there are no exposed metal parts, there's really no reason to ground it.
If there are exposed metal parts, you need to insulate them from either the electrical wiring or the user somehow.