Some random thoughts:
1) I recommend that you discuss this with a professor who knows you well and show him drafts of your statement.
2) There is no reason why you need to submit the same research statement to every school. You can focus on different research topics, depending on the strengths of each department.
3) Your list of interests above is way too long and broad. I doubt it will be taken seriously. Focus on only one or two and discuss them in enough depth to show that you really know more than just the terminology.
4) I agree with everybody else that the statement should be no longer than two pages, no matter what.
5) Nobody expects an undergraduate to have much breadth or depth in their knowledge of mathematics. What you want to demonstrate is your desire and commitment to building greater depth in your knowledge of mathematics. Although you don't want to appear too narrow (and this does not seem to be a problem for you anyway), demonstrating breadth or an interest in breadth is far less important than showing the desire for depth.
This is not a complete answer in any sense, but I will make a few comments. The irreducible subgroups $G$ of ${\rm GL}(2,\mathbb{C})$ are the primitive ones, which have $G/Z(G)$ isomorphic to $A_{4},S_{4}$ or $A_{5}$, and imprimitive groups, which have an Abelian normal subgroup of index $2.$ On the other hand, any finite group with an Abelian normal subgroup of index $2$ has all its irreducible representations of degree $1$ or $2,$ so the number of $2$-dimensional irreducible representations is easily calculated.
A more careful analysis of the primitive case shows that if $G$ has a faithful $2$-dimensional primitive complex representations, then (as has been known since the late 19th century),
$G = Z(G)E,$ where $ E \cong {\rm SL}(2,3), {\rm GL}(2,3),{\rm SL}(2,5)$ or the binary octahedral group (also, a double cover of order $48$ of $S_{4},$ (as ${\rm GL}(2,3)$ is), but with a generalized quaternion Sylow $2$-subgroup).
Now let $G$ be any finite group, and let $K$ be the intersection of the kernels of the irreducble complex representation of $G$ of degree at most $2$. The above discussion means that the only possible non-Abelian composition factor of $G/K$ is $A_{5}$, though it may be repeated if it appears. The answer to your question only depends on the structure of $G/K,$ so we may reduce to the case that all composition factors of $G$ are cyclic or $A_{5}.$ Also, by Clifford theory, we may suppose that the Fitting subgroup $F(G)$ is a direct product of an Abelian group of odd order and a $2$-group, and that all components of $G$ (if there are any) are isomorphic to ${\rm SL}(2,5).$
Further analysis can be carried out, but I believe that the analysis is not entirely straightforward. Perhaps this outline will help others to complete it, so I submit it.
Best Answer
An example from Kirillov's book on representation theory: write numbers 1,2,3,4,5,6 on the faces of a cube, and keep replacing (simultaneously) each number by the average of its neighbours. Describe (approximately) the numbers on the faces after many iterations.
Another example I like to use in the beginning of a group reps course: write down the multiplication table in a finite group, and think of it as of a square matrix whose entries are formal variables corresponding to elements of the group. Then the determinant of this matrix is a polynomial in these variables. Describe its decomposition into irreducibles. This question, which Frobenius was asked by Dedekind, lead him to invention of group characters.
A function in two variables can be uniquely decomposed as a sum of a symmetric and antisymmetric (skew-symmetric) function. What happens for three and more variables - what types of symmetries do exist there?