For your first question the answer is yes, but probably you need to look at specific examples worked out using Kazhdan-Lusztig theory. It should be enough to look at type $A_3$ (as BGG did), where values $>1$ start to appear for some K-L polynomials evaluated at 1. I believe the hand computations by BGG were quicker for an irregular weight, but that's not essential. [As you've recognized, their notation for highest weights incorporates the $\rho$-shift in a sometimes confusing way. Similarly, I've tended to use "regular" in the BGG category as shorthand for "dot-regular" relative to the shifted action of the Weyl group. But that doesn't affect your questions.]
Concerning your second question, it seems (from K-L theory) that a simple description of $N(\lambda)$ occurs mainly when $\lambda$ is dominant integral (regular doesn't matter then). See for instance section 2.6 in my book on the BGG category (AMS, 2008), for the case of a dominant integral weight. 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.