The goal of my answer is only to provide recent references.
I warmly recommend these two bits of T. Y Lam's book :
- §I.8, for examples where transvections fail to generate $SL_n(R)$
- the second to last paragraph of §VIII.12 for other interesting examples of rings $R$ satisfying $SL_2(R) = E_2(R)$ or its negative.
And also B. Magurn's latest article on generalized Euclidean group rings .
Here are newer references focussing on the instances of $SL_2(R) \neq E_2(R)$ for $R$ a quadratic order in a totally imaginary quadratic field. The state of the arts is to be found in  and , while  gives a nice geometric insight on the set $SL_2(R)/E_2(R)$.
An older, but in my humble opinion, important paper is , where the structure of $SL_2(R)$ as an amalgamated product with factor $E_2(R)$ is described for $R$ the ring of integers of a totally imaginary quadratic field (with few exceptions), see Theorem 2.4.
 C. Frohman and B. Fine, "Some amalgam structure for Bianchi groups", 1988.
 T. Lam, "Serre's problem on projective modules", 2006.
 B. Nica, "The unreasonable slightness of $E_2(R)$ over imaginary quadratic rings", 2011.
 B. Magurn, "On a note from Oliver concerning generalized Euclidean group rings", 2014.
 K. Stange, "Visualizing the Arithmetic of Imaginary Quadratic Fields", 2017.
 A. Sheydvasser, "A Corrigendum to Unreasonable Slightness", 2017.