@AliTaghavi: Yes, I mean the standard coalgebra structure on the symmetric algebra -- i.e., the one that forms a bialgebra together with the standard algebra structure, and in which bialgebra the elements of $V$ are primitive.

How much partial information do we know? Do I understand correctly that your data is of the form "the points $a$ and $b$ lie on the same side / on different sides of the chord $cd$" for various quadruples of points $a,b,c,d$ on the circle? Are you interested in results of the form "if we have $x$ many such statements, then we can uniquely recover the relative cyclic order of the points" or "if we have this specific set of statements, then we can uniquely recover..."?

The proposition also follows easily from PBW, right? (PBW yields a basis of $U\left(\mathfrak g\right)$ that includes a basis of $U\left(\mathfrak h\right)$ as a subset.)

Note that the first exact sequence here holds for arbitrary modules over any commutative ring, as long as the first nonzero term is understood appropriately (as a sum of canonical images rather than of the tensor products themselves).

Also I expect a lot of activity to follow in the anticommutative and noncommutative settings (which lag behind the commutative one for historical reasons).

There is a recent book by De Concini and Procesi with some fairly recent results in it. Usually the question is to find generators and relations for the invariant rings of a family of groups acting on a family of spaces (e.g., $\operatorname{GL}_n$ acting on $K^{n\times n}$ by conjugation). This is not a computational problem, since the families are typically infinite (all $n$, not just a few). Also, fields of positive characteristic offer a challenge in many situation where the characteristic-$0$ theory can be derived from general facts.

Likewise, the rank of $B_N$ is the dimension of the span of the matrices $P - P^T$, where $P$ ranges over the permutation matrices. Knowing a basis of the span of all permutation matrices, you can easily find a basis of the new span as well.

The rank of $A_N$ is the dimension of its column space, but its column space is the span of all permutation matrices (vectorized as vectors of length $N^2$). The latter span is known to have dimension $\left(N-1\right)^2 + 1$ (see, e.g., math.stackexchange.com/questions/70569/… ).

Updating a dead link: Proute's thesis seems to be at tac.mta.ca/tac/reprints/articles/21/tr21abs.html now (and his website at web.archive.org/web/20221010025423/http://163.172.10.123:808‌​0 , just in case).

@JosephVanName: The extra $2$ factor in your formula for $t_n$ is spurious -- I assume you have replaced the $1$s in the first column by $b_i^0 + c_i^0 = 2$'s, which caused the determinant to get scaled by $2$.

@ZachH: But the flag conditions are on columns before the transposition, aren't they? The flagged $e$-Jacobi-Trudi formula is column-flagged.

I think both the Jacobi-Trudi formula and the tableaux formulas for $\Pi$ in the original post (in its current version) are correct. Where exactly do you see an inconsistency?