What is it?

[βαθμιαίος]

1) How do you know you are a wanderer?

Unless you have dreams or visions of previous lives elsewhere, I don't think you know, but you may certainly suspect.

(and why would it matter?)

Personally, I'm not convinced it does.

2) Exactly what constitutes 'awake'.

It might be helpful to think of awakeness as a scale from "not at all" to "aware of self as Creator at all times and able to bring that awareness into manifestation at will". Surely most of us are somewhere along that scale, probably different places at different times.