[EDIT] Maybe it's useful after all this time to give a more complete and uniform answer to both of the questions asked, by referring to Theorem 3.4(i) in Springer's 1974 paper on regular elements of finite reflection groups here. In your situation, a Weyl group or other finite Coxeter group (= finite real reflection group) acts on a real euclidean space, which can be complexified to deal with eigenvalues. Springer denotes an arbitrary complex reflection group by $G$, which includes your groups.
The answer "yes" to Question 1 (or similarly the answer to Question 2) is clear from the classification of finite Coxeter groups (which include the Weyl groups along with non-crystallographic groups of type $ I_2(m), H_3, H_4$ of ranks 2, 3, 4). The numbers you call "fundamental invariants" are usually just called the degrees and were worked out long ago by Coxeter and others. (There is an exposition in Chapter 3 of my 1990 text on reflection groups, along with a table of degrees.) None of this concerns root systems or Lie algebras directly, just the theory of finite Coxeter groups. I've added to your tags accordingly.
But Springer's result provides a quick uniform proof without case-by-case verification. Here you consider a "parabolic subgroup" of $G$ (conjugate to a standard parabolic subgroup generated by some of the chosen simple reflections) or more generally a reflection subgroup corresponding in the Weyl group case to a pseudo-Levi subgroup of an algebraic or Lie group, e.g., $A_2$ inside $G_2$. Call the subgroup $H$ and note that the complexification of its natural reflection representation embeds naturally in that of $G$. So an eigenvalue for an element of $H$ (giving a nonzero eigenspace) is automatically an eigenvalue for the same element in $G$. Any such eigenvalue is a $d$th root of 1 for some $d \geq 1$.
Now Springer shows (applying some fairly elementary algebraic geometry to the hypersurfaces defined by basic invariant polynomials) that a given $d$th root $\zeta$ occurs as an eigenvalue for some element of $G$ if and only if $d$ divides one of the degrees of $G$.