Symplectic Steinberg group

Here's a small follow-up on Matsumoto's thesis, which deals essentially with the congruence subgroup problem for Chevalley (split) algebraic groups. This followed work by Bass-Milnor-Serre, but in turn was followed by more technical work on nonsplit groups (Prasad-Raghunathan, in particular). In my 1980 Springer Lecture Notes 789 on Arithmetic Groups, I tried in the last part to convey Matsumoto's ideas in the special case of $SL_n$ when $n \geq 3$ but with some side remarks about the general case. The congruence subgroup problem has a different solution for $n=2$, which should for this and some other purposes be assigned to type $C_1$ rather than the conventional $A_1$. For symplectic groups there is a significant difference, mentioned in my Remark on page 129. Briefly:

The ideas in my $SL_n$ proof carry over almost unchanged to other Chevalley groups of rank $\geq 2$, with one important modification due to the fact that real Lie groups of type $C$ starting with rank 1 have an infinite cyclic fundamental group while others have a finite fundamental group. This is what complicates the proof for symplectic groups in Matsumoto's paper (which is only available in French). The special feature of root systems of type $C$ seems to be the existence of roots equal to twice a weight.

There is useful information about the symplectic analogues of the Steinberg group, the Steinberg symbols, and the $K_2$ functor in some of Michael Stein's papers from the '70's. In particular, see his papers "Generators, relations and coverings of Chevalley groups over commutative rings" and "Surjective stability in dimension $0$ for $K_{2}$ and related functors" and "Injective stability for $K_{2}$ of local rings".

There is an excellent survey paper: Linear Algebraic Groups and K-theory http://users.ictp.it/~pub_off/lectures/lns023/Rehmann/Rehmann.pdf by Ulf Rehmann. It seems that Matsumoto paper concerns symplectic case, that is the answers to all your questions are positive.

Note that for non-symplectic groups, Steinberg symbols are bilinear, however for symplectic it is not true. For a nice description of the 2-cocycle of the topological universal cover $\widetilde{SL_2({\mathbb R})}$ of $SL_2({\mathbb R})$ see e.g. Asai, T.: The reciprocity of Dedekind sums and the factor set for the universal covering group of $SL(2,{\mathbb R})$.

From Asai work, one can deduce that the Steinberg Symbol corresponding to a $\widetilde{SL_2({\mathbb R})}$ is defined as: for $x,y\in {\mathbb R}^{\times}$

$c(x,y) = \left\{ \begin{array}{l l} -1 & \quad \text{if } x < 0 \text{ and } y<0 \\ 0 & \quad \text{otherwise}\\ \end{array} \right.$