The answer "yes" to Question 1 looks 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 real reflection groups (= finite Coxeter groups). I've added to your tags accordingly.

Even more is true: the non-crystallographic groups are embedded in certain Weyl groups of type $A,D,E$ and twice the rank by a subtle "folding" procedure explained carefully in the 1988 paper of Oleg Shcherbak referenced in my book (which is apparently based on the Ph.D. thesis he was working on as V. Arnold's student in singularity theory at the time of his death in about 1986). Here the degrees occur in fact among those of the larger rank group involved, not just as divisors of those degrees. The same is true (using the classification) for groups obtained more conventionally from Weyl groups with a single root length by "folding" their Dynkin diagrams, such as $F_4$ obtained from $E_6$. In all such cases the two groups share the same Coxeter number, which equals the largest degree of a homogenous invariant polynomial among the generators of the algebra of $W$-invariants.

Having observed this much empirically using the classification, it is still a problem to approach your Question 2. For this you'd need to look at the actual polynomial invariants, preferably from a uniform viewpoint. In the individual cases you might see some natural relationship between "basic" invariant polynomials: homogeneous ones generating the invariant algebra, whose degrees are uniquely determined and have the group order as product. I haven't noticed relevant work in the literature, though that is quite scattered and involves some papers in the mathematical physics journals. Some references are indicated in the notes at the end of my Chapter 3, though I overlooked one thesis-related paper by Lee here. (This and related matters are discussed in some earlier questions at Math Overflow in connection with "polynomial invariants of finite reflection groups".)

In the parallel case of "folded" Dynkin diagrams, the non-crystallographic groups of type $H_3$ were studied in the 1979 paper by Sekiguchi and Yano here from the viewpoint of their embedding of $H_3$ into a Weyl group of type $D_6$. In great detail, but with little verbal explanation, they explore the polynomial invariants of the two groups. Essentially what seems to happen here is that after identifying pairs of vertices of the Coxeter graph appropriately (by "folding"), they can obtain 3 basic invariant polynomials from 6 for the larger group.

In your situation, where for example in type $A_n$ one symmetric group (or product of such) is embedded in a larger symmetric group, it may be possible to relate the symmetric polynomials in a natural way. More generally, as suggested in the comments, similar arguments should apply to "pseudo-Levi" types obtained by omitting one or more vertices from an extended Dynkin diagram. Again the degrees always divide. But I'm not sure how to carry out such a program, even case-by-case.

[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 $G$.