Deep strategical analysis certainly fits within the realm of "Board and Card Games" but these questions and their resulting discussions tend to drift into a different style of Q/A. Is there an unofficial line that defines how far into this realm we should be going?
Topics I consider relevant to this question:
Data gathering/mining (Where/how do I get X?)
Turn/game format discussion (What/how should I store X?, What turn formats are available for X?)
Probability (What are the odds of X occurring?)
Player strength terminology/format (ELO, TrueSkill)
Specific strategies (How do I simulate opening X?)
Specific games (How do I simulate games for X?)
AI development (How do I create an AI that plays X?)
AI competition (Where can my AI compete at X?)
Help with specific AI (How do I beat X?, How do I get X to work?)
General AI discussion (What type of AI would be best for X?)
And so on. The main question here: Are people interested in analyzing game-theory, AI and the mathematics of board games encouraged to post those questions here? Or in a more relevant area of stackexchange? Or... ?
All the topics you mention in your question sound potentially interesting to me. I think questions on all of those should be allowed/encouraged, but if particular questions are very [maths/programming/stats] oriented then it would be worth suggesting the questioner also posts on the relevant stack exchange site.
FWIW, I am exceptionally happy to see quantitative questions and responses here. Of course, having the ability to cross post a thread about boardgames and statistics to both a boardgames and a statistics stackexchange would be very interesting (to me).
I do play board and card games for the social aspect but I cannot turn off that part of my mind that loves AI. My primary question during and after a game is: "How must an AI be designed to solve the problems involved". I also love to find simple algorithms that I can use to find better solutions during the game.
So yes, I would love to see more discussion about the mathematical aspects, algorithms and AI.