Griffiths and Harris (Principles of Algebraic Geometry) has a nice introduction to Schubert calculus, and they explicitly work out their general formulas for the case $k=2, n=4$, so this might be a particularly useful resource for the OP.

I think there's a strong argument to be made that in an intro course (esp. with undergrads whose alg. background may be limited), that it's a good idea to only prove that the image is closed. One could supplement that with an explanation of why algebraic geometers are interested in proving the stronger statement that the map is a closed immersion (using the manifold analogy to motivate why) and then mention that in this case one has to work over rings, not just alg. closed fields, and the theory to do this is the theory of schemes, something you should learn in your next alg. geom. course.

I thought Chevalley's paper also works over $\mathbb{C}$. Have you tried looking in Procesi's book Lie Groups: An Approach Through Invariants and Representations

None of those properties characterize toric varieties (i.e. there are non-toric varieties that satisfy them). You can revise (1) to get an exact criterion: $X$ is a toric variety iff $X$ is a Mori Dream Space and $\text{Cox}(X)$ is a polynomial ring.

Sadly, the links are no longer active. It is such a great article that, with Dr. Howe's permission, it would wonderful if it had a permanent online home.

The classical topology on projective space does not induce the Zariski topology on the Grassmannian; it induces the classical topology on the Grassmannian. Many topological statements about the Grassmannian are true for both choices of topology because in practice, most of arguments involving open sets in the classical topology can be made using only open sets from the Zariski topology.

A perfectly reasonable approach. I don't know enough about $\mathbb{F}_1$ to have any idea what approach is more tractable. It is certainly possible that an analogue of integrating against a differential form is easier to generalize. Still the non-simply connectedness of $\mathbb{C}^*$ and the impossibility of defining an entire logarithm in the case of $k = \mathbb{C}$ makes me think that the "topology of $\mathbb{F}_1^*$" might be quite complex.

The exponential function is generally easier to understand than than the logarithm, since it is analytic on all of $\mathbb{C}$ and satisfies a simple first-order ODE. Typically, the logarithm is then defined as the inverse of the exponential function. So it seems natural to first try to understand the "exponential function over $\mathbb{F}_1$" (whatever that might mean!) and then base your understanding of the "logarithm over $\mathbb{F}_1$ off of that.

The literature on this topic is vast, but one place to start would be papers by N. Bergeron and Sottile. They solved the Schubert times Schur problem in terms of chains in $k$-Bruhat order. The $k$-Bruhat order is generated by covering relations in ordinary Bruhat order $u \prec v = u \cdot (a,b)$, $\ell(v) = \ell(u) + 1$, with $a \leq k < b$.