How do I get Cubase to automatically reset held channel peak levels on playback?
You can’t. As a workaround, you can set a keycommand to reset them with a key. Same for the performance meter.
How do I reset the peak level number display without having to click on it for a specific channel?
You can change the Peak Hold Time in your preferences.
Add reset and start/stop to a macro then set it the space bar.
Just to be clear, it’s this number I’m referring to.
I don’t necessarily want to reset all channels, just tje selected one.
I can do it with a macro but the problem there is it will reset the number on stop, too.
The other way would mean using two different keys for start and stop (I think they are numeric keypad Enter and 0 by default) but I don’t want to do that, I want to keep the toggle.
I thought you’d say that. 
If you really need the toggling behaviour, you can create 2 key command preset files, one that has the reset->start/stop->select KC set 2, and the other has start/stop->select KC set 1. Keypad 0 also should flip the state and you’ll have to remember not to use the mouse to stop it. That’s too troublesome I guess?
I have the key command assigned to Ctrl Alt NumpadEnter and use it often. I don’t even let go of the mouse, I just pop my thumb over to the Enter key.
But how do I reset the peak level number display (without having to click on it) just for a specific channel?
I don’t think you can reset a specific channel with a key command. I’m away from my Cubase at the moment so I can’t remember if the Reset Peaks in the MIDI Remote could do this for just the currently selected channel, but then again, you didn’t mention anything about using a MIDI controller, so I’m not sure it will suit you.
I’m just looking for a keyboard shortcut to avoid the need for clicking on the number on the specific channel before each playback.
I see. But is there a specific reason why you need just the peak of a specific channel to be reset? Why not all at once?