No. For each countable ordinal $\alpha$, the bounded Baire class $\leq \alpha$ functions on $[0,1]$ constitute a C-algebra. These C-algebras are nested, all are contained in $L^\infty[0,1]$, and every semicontinuous function is already in Baire class one.

See http://www.encyclopediaofmath.org/index.php/Baire_classes

No. The bounded Baire class one functions on $[0,1]$ are stable under uniform limits and hence constitute a C-algebra. This C-algebra contains every semicontinuous function on [0,1]. Every function in $L^\infty[0,1]$ is equal almost everywhere to a Baire class two function, but not a Baire class one function. (The previous version of my answer neglected this essential point.)

See http://www.encyclopediaofmath.org/index.php/Baire_classes