1 - the control channel is always TS1 on whatever frequency is being used. It can rotate to other channels so you have to be careful about leaving slots out. I used to leave out the slot that the CC was using but then realized I was missing traffic (when watching the system with DSD+).
2 - I have no idea what an LUN is, sorry. That's not something I've seen before related to DMR.
3 - yes, the radio ID is on the top line during a call. The radio will always show the active talkgroup during a transmission, not the channel name. You cannot override this.