I think the .005 - .010 dif. is coming from backlash. When you home it, it goes till it sees the switch then backs off untill it doesn't. If you tell the machine to go to 0, it is approaching from the opposite direction.
1) I think G28.1 will do this
2) I'm not sure.
3) G53 Z0. is what you want I think.
4) Right beside the stop button on the program run page you should see a line number. This is a DRO. You can type the line number in (click in the box, type the number, hit enter) and click run from here. Even if your program doesn't have line numbers, you can count them or see what line it is on when you stop it. If you want to move it then pick back up on that line, type it in Line number and click run from here.
5) I think you are talking about the MPG screen. Hit your tab key to expand and collapse it.
Brett