Try a one sided zigzag bicomplex $$K^{p,q} = \begin{cases} \mathbb{Z}, & 0\leq q=-p \text{ or } 0\leq q=-p+1; \\ 0, & \text{otherwise.} \end{cases}$$ with the differentials $d_v^{p,q}$, $d_h^{p,q}$ equal to the identity when source and target are $\mathbb{Z}$.

The span of the vectors $(Y^p-X^{p-1}Y)^iY^{g-pi}$ is better described as the span of the $(X^{p-1})^iY^{g-pi+i}$.

No, in the linear algebraic-group setting many modules are not finite dimensional. They are comodules for the coordinate ring viewed as a Hopf algebra. Induction is not defined the way you seem to think. See the book Representations of Algebraic Groups by Jantzen for all this.

Vincent Franjou and Wilberd van der Kallen , Power reductivity over an arbitrary base, Documenta Mathematica, Extra Volume Suslin (2010) , pp. 171-195.

In characteristic two one sees that your definition of polynomial endofunctors is not very good: In the exact sequence $0\to \wedge^2\to\otimes^2\to S^2\to 0$ the $\wedge^2$ does not satisfy your definition. The definitions by Friedlander and Suslin are better. Also note that their category is no full subcategory of $Fun(\rm Vect,Vect)$.

So maybe there should be a warning that this is not what the Lectures on functor homology are about.

It looks as if you consider any additive map hom(U,V)→hom(F(U),F(V)) to be polynomial. But we do not consider the complex conjugation map $\Bbb C\to\Bbb C$ to be polynomial. Tensoring with $M$ defines an additive functor $F$ that is not strict polynomial in the sense of Friedlander and Suslin.

But I found a counterexample in the Habilitation thesis of Antoine Touzé. It goes like this: Tensor each complex vector space with the bimodule $M=\Bbb C$ with ordinary multiplication on the right, but multiplication by complex conjugate on the left.

I did not find the proof of equivalence of the two notions in the paper of Eilenberg and MacLane.

Friedlander and Suslin distinguish between polynomial functors, defined in terms of cross effects, and strict polynomial functors, defined in terms of polynomial maps hom(U,V)→hom(F(U),F(V)).

Quillen was an excellent expositor and chose a formulation whose meaning was immediately clear to contemporary readers.