(2) Probably not suitable for an undergraduate course, but if $s_{(n-1,n-2,\dots,1)}$ denotes the Schur function of the staircase shape $(n-1,n-2,\dots,1)$, then the evaluation $$s_{(n-1,n-2,\dots,1)}(x_1,\dots,x_n)=\prod_{1\leq i \lt j\leq n} (x_i+x_j)$$ follows immediately from the bialternant formula for Schur functions, since it reduces to the quotient of two Vandermonde's: $\prod (x_i^2-x_j^2)/\prod(x_i-x_j)$. See Exercise 7.30 of Enumerative Combinatorics, vol. 2.