Thank you, DotterKat. My native speaker rejects "nagmaneho nang lasing" - insisting on "nagmaneho na lasing" on the grounds that "lasing" does not describe the manner of the action. This seems more than simply a colloquial preference for "na". However, I find support for "nang lasing" on the internet, e.g.:
Hindi siya puedeng mag-maneho nang lasing.
Pano ka nakapag-manejo nang lasing?
... so perhaps it is just that different speakers have different preferences.
Schachter & Otanes 1972 (6.7, 6.10) seem to accept "nang" only before an adverb of manner or time. In English, the Cambridge grammar analyses "drunk" in "drive drunk" as a "predicative" (or "stative") adjunct, since it describes the state of the subject while driving rather than the manner or time of his driving. In S&O the closest relevant example of "lasing" that I could find uses "na":
Inabutan ko si David na lasing = I found David drunk.
and likewise Kroeger 1973:
naghain na lasing si Maria ng isda = Maria served the fish drunk.
That said, I do see an advantage in consistently using "nang" before "lasing" because it would resolve the ambiguity of:
nagmaneho si Tom na lasing (Is it drunken Tom driving or Tom driving drunk?)
I see that your last example cleverly skirted the ambiguity by using "ay":
si Tom na lasing ("Drunk Tom") ay nagmaneho = Tom, who was drunk, drove.
Regarding (2), my native speaker agrees with the comma but still prefers "nang" or "na" after it:
[Habang] lasing, [na/nang] nagmaneho si Tom. = Drunk, Tom drove.
..regardless of the presence of "habang". Again, I do not know if this preference is idiosyncratic.