[Edited again to give a second identity relating $E$ to eta products]
Continued fraction or not, an expression $q^{\frac{(r-s)^2}{8(r+s)}} f(\pm q^r, \pm q^s)$ is a modular form of weight $1/2$ for all integers $r,s$ with $r+s>0$, because it is a sum $\sum_{n=-\infty}^\infty \pm q^{(cn+d)^2}$ with rational $c,d$ and periodic signs. Therefore the quotient of two such expressions is a modular function, and takes algebraic valus at quadratic imaginary values.
The quotient $$ E(q) = q - q^2 + q^6 - q^7 + q^8 - q^9 + q^{11} - 2q^{12} + 2q^{13} - 2q^{14} + 2q^{15} \cdots $$ looks like a modular unit $-$ its logarithmic derivative has small coefficients $-$ but not quite an eta product; instead it seems to be a quotient of Klein forms: $$ E(q) = q \prod_{n=1}^\infty (1-q^n)^{\chi(n)}, $$ where $\chi$ is the Dirichlet character of conductor $12$, given by
$$ \chi(n) = \cases{ +1,& if $n \equiv \pm 1 \bmod 12$; \cr -1,& if $n \equiv \pm 5 \bmod 12$; \cr 0,& otherwise. } $$ One identity relating $E$ to an $\eta$ product, similar to but somewhat more complicated than the ones you give for $A,B,C,D,$ is $$ \frac1{E(q)} - E(q) = \frac{\eta(2\tau)^2 \eta(6\tau)^4}{\eta(\tau)\eta(3\tau)\eta(12\tau)^4}. $$ Two identities relating $E$ to $\eta$ products, similar to but somewhat more complicated than the ones you give for $A,B,C,D,$ are $$ \frac1{E(q)} - E(q) = \frac{\eta(2\tau)^2 \eta(6\tau)^4}{\eta(\tau)\eta(3\tau)\eta(12\tau)^4}, $$ and (a bit simpler) $$ \frac1{E(q)} + E(q) = \frac{\eta(4\tau)}{\eta(\tau)} \Bigl(\frac{\eta(3\tau)}{\eta(12\tau)}\Bigr)^3. $$
