I have two specific questions. The first is the result I actually need, and the second would let me prove it.
EDIT: The second statement was wrong. I am keeping it for posterity. I am adding a third statement that reflects the minimum requirements of the proof I had in mind for the first statement.
I need to show that for $0<\epsilon<1$ and $g\in H^{\epsilon}(\mathbb{R}^2)$, $\frac{g}{r^{\epsilon/2}}\in L^2(\mathbb{R}^2)$.
The following statement was shown by responders to be false: It would help me to know whether it is true that if a compactly supported function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is Holder continuous with $|f(x)-f(y)|\le C|x-y|^{\alpha}$ for some $0<\alpha<1$ and a constant $C>0$, then $f\in H^{\frac{n}{2}+\alpha}$.
Corrected version of second statement: It would help me to know whether it is true that if $f$ is an $H_0^{1+\epsilon}(\mathbb{R}^2)$ function with $f(0)=0$, $\frac{f}{r^{\epsilon/2}}$ is in $H^1(\mathbb{R}^2)$. I don't know whether this is any easier or not, but we know that $f$ satisfies $|f(x)|<Cr^{\epsilon}$ in this case by the Sobolev imbedding theorem
I have an intuition that $g$ is an $H^{\frac{n}{2}+s}(\mathbb{R}^n)$ function, then for $0<s<1$ and $g(0)=0$, or for $s\le0$ regardless, we should have that for any $\nu>\delta>0$, $\frac{g}{r^{\nu-\delta}}$ is in $H^{\frac{n}{2}+s-\nu}(\mathbb{R}^n)$, presuming that $g(0)=0$ if $s>0$.
This is based on the idea (which may or may not be correct) that fractional derivatives of singularities should increase their order at the same rate as integer derivatives (that is, if a rapidly decaying function $f$ has a singularity of order $|x-x_0|^{\alpha}$ at $x=x_0$, I think that $(\frac{d}{dx})^\alpha(f(x))$ should have a singularity of order $|x-x_0|^{\alpha+\beta}$ at $x=x_0$. I don't know how to prove this and probably don't need to, although it would likely be sufficient to get my result.
The converse of the second bolded statement is true and is known as the Sobolev imbedding theorem (see for example Theroem 3.26 in Strongly Elliptic Systems and Boundary Integral Equations by William McLean).
If anyone knows of a proof or reference for one of the bolded statements, that would be incredibly helpful.
Counterexample to the second one: $n=4$, $f(x)=|x_1|\phi(x)$, where $\phi$ is smooth and compactly supported and $\phi(0)=1$. Then $f''_{x_1x_1}=2\phi(x)\delta(x_1)+\dots$ is not in $L^2$.