You asked a lot about the F–S indicator. I'll answer part of it for now, so that there is a smaller piece to answer later.
Your first division algebra question makes more sense in a slightly different context: if T is an irreducible real representation of G (an irreducible ℝG-module), then three cases occur: EndℝG(T) = ℝ is real, = ℂ is complex, or = ℍ is quaternionic. In the first case T is also irreducible over the complexes and its Schur indicator is +1. In the second case T is not irreducible over the complexes, but is instead the sum of two complex conjugate representations, both of whose Schur indicators are 0. In the third case, the representation "ramifies" when extended to the complexes, and you get the sum of two isomorphic representations (both of) whose Schur indicator is -1.
I like to think of this in terms of complex endomorphism rings, but then it no longer specifically mentions the quaternions: EndℂG(T) = ℂ is complex when the Schur indicator is +1, = ℂ×ℂ when the Schur indicator (of the summands) is +0, = M2(ℂ) when the Schur indicator (of the summand) is -1. These correspond to applying –⊗ℝℂ to the reals, complexes, and quaternions.
Your numerical invariant of the division algebra is probably best understood as a quest for the "Brauer group" of F. There is a very well-defined and reasonably well-understood theory for this. That is usually phrased in terms of "central simple algebras", but it turns out to mostly be a matter of notation. For F a p-adic field, then yes, you mostly just get a number, but for a general field it is a little bit of a stretch (but not insane) to say you get a number.
On the off chance someone is interested, this is one of the places where the Schur index is easily understood. If the Schur indicator is +1, then the field of values and the splitting field are equal to the reals. If the Schur indicator is 0, then the field of values and the splitting field are equal to the complexes. If the Schur indicator is -1, then the field of values is the reals, but the splitting field is degree 2 (the complexes), so the Schur index is 2. This is also the smallest multiple of the character that has a representation realizable over the field of values, and so this is is why in the -1 case, T splits into two (self-conjugate) copies of itself. In terms of blocks of the group ring, this block is a matrix ring over the quaternion algebra, and it contains many copies of the irreducible. Each copy of the irreducible has a centralizer isomorphic to the complexes, a splitting field of the block and the irreducible.
Edit: A gentle book for character theory is G. James and M. Liebeck's Representations and Characters of Groups. Its chapter 23 is entirely to devoted to all of these different ideas (as well as Frobenius's original motivation) without using anything fancy. It has many examples, many exercises, and solutions to many exercises. I think it is useful for both group theorists and people who just want to use representations.
For fancier things: I. Reiner's Maximal Orders has nice coverage of the division algebras associated to finite group algebras (and their maximal orders). My favorite textbook treatment of the numerical invariant of the division algebras are Albert's Modern Higher Algebra and Structure of Algebras, but some people think they are old fashioned. Recent treatments will often get lost in technicalities that are irrelevant to group algebras. You might be OK with one of large textbooks on algebra; many have a chapter on Central Simple Algebras and the Brauer Group. MathOverflow readers will probably like Rowen's Algebra: A Non-commutative View.
As pointed out in comments and at this question, this answer is not entirely correct because real/quaternionic does not always give a $\mathbb{Z}/2$ grading. Nonetheless in practice this answer seems to be "mostly right." I'm going to leave it up unmodified because changing it would confuse the comments and other answers too much. Sorry for the mistake!
If $G$ has a quaternionic representation and $G$ has no complex representations (i.e. reps whose character is complex) then for certain nontrivial central elements the square root function is $\sum_\chi \chi(1)$ which is clearly the maximum possible value and clearly larger than the value on the identity.
The key idea in the proof is the fact that $Z(G)^*$ (the group of characters of the center of $G$) is the universal grading group for the category of $G$ representations.
Let me unpack that a little bit. To every irrep of $G$ you can assign it's "central character", that is an element of $Z(G)^*$. This assignment is multiplicative in the sense if $U$ is a subrep of $V \otimes W$ then the central character of $U$ is the product of the central characters of $V$ and $W$ (this is just because central elements act by scalars). In other words, $\operatorname{Rep}(G)$ is graded by $Z(G)^*$. The claim is that an $H$-grading of $\operatorname{Rep}(G)$ is the same thing as a map $Z(G)^* \to H$.
Why is this true? I'm just going to sketch the proof, in particular I'm just going to talk about the case where $Z(G)$ is trivial, but I'll indicate how the general case works. Since the center is trivial you can find a faithful representation $V$. But then let's look at a really high tensor power of $V$ and ask how it breaks up. We can compute that using character theory. Since the rep is faithful and there's no center, the character of a high tensor power of $V$ is dominated by the value on $1$. Hence any high tensor power of $V$ contains all other irreps. In particular, the $n$-th and $(n+1)$-th powers contain the same irreps and this tells you that the grading group is trivial. In general you want to argue that the contribution of the center dominates the values of inner products (but you need to be a bit careful about non-faithful representations).
The "universal grading group" is used by Gelaki and Nikshych to define the upper central series of an arbitrary fusion category and thereby define nilpotent fusion categories.
Ok, how is this relevant to anything? Well, if you have a group with a quaternionic representation but no complex representations, then the Frobenius-Schur indicator gives a $\pm 1$ grading of $Rep(G)$, namely put the real reps in grade $1$, and the quaternionic ones in grade $-1$. Hence the Frobenius-Schur indicator $s(\chi)$ must be given by the central character $\chi(z)/\chi(1)$ for some central element $z$. Clearly these central elements maximize the square root function.
Best Answer
Here are some things you probably know. For a representation $W$ of $G$, let $\text{Inv}(W)$ denote the subspace of $G$-invariants. For an irreducible representation $V$ with character $\chi$, the F-S indicator $s_2(\chi)$ naturally appears in the formulas
$$\dim \text{Inv}(S^2(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^2 + \chi(g^2)}{2}$$
and
$$\dim \text{Inv}(\Lambda^2(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^2 - \chi(g^2)}{2}.$$
More precisely the F-S indicator is their difference, while their sum is $1$ if $V$ is self-dual and $0$ otherwise. The corresponding formulas involving $s_3(\chi)$ are
$$\dim \text{Inv}(S^3(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^3 + 3 \chi(g^2) \chi(g) + 2 \chi(g^3)}{6}$$
and
$$\dim \text{Inv}(\Lambda^3(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^3 - 3 \chi(g^2) \chi(g) + 2 \chi(g^3)}{6}.$$
Here the F-S indicator $s_3(\chi)$ naturally appears in the sum, not the difference, of these two dimensions. $T^3(V)$ decomposes into three pieces, and the third piece is (Edit, 9/26/20: two copies of) the Schur functor $S^{(2,1)}(V)$, which therefore satisfies
$$\dim \text{Inv}(S^{(2,1)}(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{ \chi(g)^3 - \chi(g^3)}{3}.$$
So $s_3(\chi)$ constrains the dimensions of these spaces in some more mysterious way than $s_2(\chi)$ does. The sum
$$\dim \text{Inv}(T^3(V)) = \frac{1}{|G|} \sum_{g \in G} \chi(g)^3$$
tell us whether $V$ admits a "self-triality," and this dimension is an upper bound on $s_3(\chi)$. If $V$ is self-dual, this is equivalent to asking whether there is an equivariant bilinear map $V \times V \to V$, which might be of interest to somebody. If this dimension is nonzero then $s_3(\chi)$ gives us information about how a triality behaves under permutation.
The situation for higher values of $3$ is worse in the sense that the bulk of the corresponding formulas are not completely in terms of F-S indicators but in terms of inner products of F-S indicators and their interpretation will only get more confusing. Already I don't know of many applications of triality (in fact I know exactly one: http://math.ucr.edu/home/baez/octonions/node7.html).