The answer to this question is yes. Here is an overkill solution:

[fixing ...]

Q1 has a negative answer.

Let $X$ be an infinite Turing-independent set and fix elements $(r_i)_{i\in\omega}\in X$. The real $\bigoplus_{i\in\omega}r_i$ cannot be computed from the join of any finite $F\subseteq X$, since the join of such an $F$ would then compute some $r_i\not\in F$.

Of course, this does not address Q2.