For your first question the answer is no.  See for instance section 2.6 in my book on the BGG category (AMS. 2008).  This relies mainly on the early work of Harish-Chandra but was re-emphasized by Verma in his 1966 thesis and came to play a major role in the development of the BGG resolution.   On the other hand, Kazhdan-Lusztig theory suggests strongly that there is no reasonable condition for non-dominant weights.  (Note that regularity is not required for dominant integal weights.)