An answer to (2) in ZFC:  The algebra $\mathcal B$ is not only uncountable, it  has the property that below any positive element there are uncountably many elements. 

Let  $\mathcal A'$ be the algebra of all sets of the form $([a_{0},b_{0})\cup[a_{1},b_{1})\cup...\cup[a_{n},b_{n}))\cap I$ where all $a_i,b_i$ are in $([0,\frac12]\cap \mathbb Q) \cup [\frac12,1]$.  This is a subalgebra of $\mathcal A$.  So $\mathcal A'/N$ is a subalgebra of $\mathcal A/N$, it is uncountable and atomless.

Now consider the element $[0,\frac12]/N$ in $\mathcal A'/N$.  It is positive, but there are only countably many elements below it. 

So  $\mathcal A'/N$ is not isomorphic to $\mathcal B$.