Are open $\mathbb{G}_m$-invariant subschemes of an affine scheme precisely the homogeneous radical ideals of its coordinate ring?

Your definition

$\mathbb G_m \times Y \hookrightarrow \mathbb G_m \times X \to X$ factors over $Y \hookrightarrow X$

can be simplified a bit. A better one for our purposes is

The two subschemes of $\mathbb G_m \times X$ defined as the inverse image of $Y$ under the right projection $\mathbb G_m \times X \to X$ and the action map $\mathbb G_m \times X \to X$ are equal.

To see they are equivalent, note that the universal property of a fiber product implies your factorization is equivalent to the claim that the immersion $\mathbb G_m \times Y \hookrightarrow \mathbb G_m \times X $ factors through the map from $\left(\mathbb G_m \times X \right) \times_X Y $ to $\mathbb G_m \times X $. That fiber product is the inverse image of $Y$ under the action map, and $\mathbb G_m \times Y$ is the inverse image of $Y$ under the right projection. So your factorization is equivalent to $p^* Y \subseteq a^* Y$, where $p$ is projection and $a$ is action. But $p^* Y \subseteq a^* Y$ is equivalent to $a^* Y \subseteq p^* Y$, and therefore equivalent to $p^* Y =a^* Y$, because we can swap the projection and action maps using the automorphism of $\mathbb G_m \times X$ that sends $(g,x)$ to $(g^{-1}, gx)$.

Now this one is equivalent for open subschemes and their closed complements because the operation of taking closed complement is compatible with pullback under smooth morphisms.

Your conjecture 2 is false if you don't assume that $G$ is reduced (in positive characteristic there are affine group schemes that are not reduced).

As to conjecture 3, it is hopelessly wrong (think of the action of $\mathrm{GL}_n$ on $\mathbb P^{n-1}$). The only non-trivial case I know is Sumihiro's theorem: a normal algebraic variety with an action of a torus can be covered by invariant affine open subsets. Being normal is essential: consider the standard action of $\mathbb{G}_{\mathrm m}$ on $\mathbb P^1$, and glue together the origin and the point at infinity.