The answer to question 1 is yes. Take the binary binary branching tree $2^{<\omega}$ and label every node with a different natural number. Now, for each branch $b$ through the tree, let $X_b$ be the set of labels on those nodes. Distinct branches $b$ will give continuum many set $X_b$, and none of these is a subset of another. Indeed, they are an *almost-disjoint family* of size continuum, a family of infinite sets, with any two having finite intersection.

For question 2, the answer is negative in ZFC, since we would be assigning to each element of the family a distinct element of the set, and so it couldn't be larger.

But in ZF, if we replace "$|S|=|P(X)|$" with "$|S|>|X|$", then it can happen (with the subset characterization, rather than element characterization, since without AC you can't pick the element). Namely, it is known to be relatively consistent with ZF that there is a partition $S$ of the real numbers $X=\mathbb{R}$ into *strictly more* than $\mathbb{R}$ many elements. Basically, if there is no $\omega_1$-sequence of reals, then we can define an equivalence relation on $\mathbb{R}$ by the coding-the-same-countable-ordinal relation, where two reals are equivalent if they both code the same countable ordinal and otherwise inequivalent, unless identical. Now, we get at least $\omega_1$ many equivalent classes, plus continuum many more, and so the number of partition elements exceeds $\mathbb{R}$, but they are disjoint.

Meanwhile, if you insist on $|S|=|P(X)|$, then I can prove it is impossible in ZF. By picking for each element of $S$ the largest subset of it that is disjoint from all the other elements of $S$, we can thereby produce a partition of $X$ of size $P(X)$. This is precisely the situation I mentioned in the comments, which I can now rule out.

Specifically, suppose that $S$ is a partition of a set $X$ and $S$ has size $P(X)$. So to each subset $A\subseteq X$, we have a distinct element $X_A\in S$. Define a map $f:X\to P(X)$ by $f(x)=A$, when $x\in X_A$. Since $S$ is a partition, this defines a function. And our assumptions ensure that this is surjective. But there is no surjection of a set $X$ to $P(X)$ by Cantor's argument.

So the answer to question 2 is negative.