A fairly trivial sufficient condition for the existence of a complete Boolean algebra of projections is for the space $X$ to be (up to isomorphism) of the form $(\sum_{n\in\mathbb{N}} \oplus E_n)_{\ell_p}$, 1\leq p < \infty, or $(\sum_{n\in\mathbb{N}} \oplus E_n)_{c_0}$ (where the spaces $E_n$ are, of course, nonzero).
So, an example of a space with the property requested in the OP's second question is an space of the form $c_0(E)$ or $\ell_p(E)$, 1\leq p < \infty, where $E$ is a separable Banach space that does not embed in any Banach space having an unconditional basis; examples of such spaces $E$ include:
The James space $J$.
The James tree space $JT$.
$L_1[0,1]$.
Let $K$ be either an uncountable compact metric space (e.g., $[0,1]$), or equal to a compact ordinal interval $[0,\alpha]$, where $\alpha \geq \omega^\omega$ and $[0,\alpha]$ is equipped with its natural order topology. Then one can take $E=C(K)$.
The universal basis space $U$ of Pelczynski, which has (a basis and) the property that every Banach space with a basis is isomorphic to a complemented subspace of $U$. In particular, $U$ does not embed in a space with an unconditional basis since it contains (complemented) copies of $L_1[0,1]$ and the $C(K)$ spaces mentioned above (since these spaces all have a basis).
With the exception of the James space, all of the examples $E$ given above have the property that $E$ is isomorphic to $c_0(E)$ or $\ell_p(E)$ for some 1\leq p < \infty (in the case of Pelczynski's space, we actually have that $U$ is isomorphic to all of the spaces $c_0(U)$ and $\ell_p(U)$, 1\leq p < \infty); so, with the exception of the James space, we can in fact take any of the spaces $E$ given as examples above as an answer to the second question.
Finally, I mention that one could also consider, in a similar way, direct sums with respect to any unconditional basis.