2. Diēs is typically masculine (like most fifth declension nouns), but is occasionally feminine, especially in fixed phrases and general reference to time or dates.
cōnstitūtā diē : on a fixed day
longa diēs intervēnit : a long time had passed
3. Diēs is one of only two nouns in the fifth declension that is entirely declined. Rēs is the other such noun—all other fifth declension nouns are wanting in the plural (or at least the plural genitive, dative and ablative) in extant Latin literature.
There are a number of fixed phrases in Latin that are accusative idiomatically. Here are a few:
Trēs hominēs id temporis exībam: I was dating three guys at that time.
Puellī molestiōrēs id aetātis fuerunt: Boys are a hassle at that age.
Quod genus anī est: What sort of old woman is she?
Meam vicem, habēas sicut placet: As far as I am concerned, you may have as much as you like.
Quid craterā hoc noctis agis: What are you doing with a mixing bowl at this hour of the night?
It isn’t clear to me how Allen and Greenough identify ‘quod (or id) genus‘ as an adverbial accusative. They agree it may have been a nominative at one point, but how do we know it isn’t a nominative now? Is it simply because ‘adverbial nominative’ isn’t an extant class?