On the existence of almost disjoint and MAD families without $\mathsf {AC}$

In set theory without the Axiom of Choice ($\mathsf{AC}$), we investigate the deductive strength and mutual relationships of the following statements:

(1) Every infinite set $X$ has an almost disjoint family $\mathcal{A}$ of infinite subsets of $X$ with $|\mathcal{A}|\not\leq\aleph_{0}$.

(2) Every infinite set $X$ has an almost disjoint family $\mathcal{A}$ of infinite subsets of $X$ with $|\mathcal{A}| \gt \aleph_{0}$.

(3) For every infinite set $X$, every almost disjoint family in $X$ can be extended to a maximal almost disjoint family in $X$.

(4) For every infinite set $X$, no infinite maximal almost disjoint family in $X$ has cardinality $\aleph_{0}$.

(5) For every infinite set $A$, there is a continuum sized almost disjoint family $\mathcal{A}\subseteq A^{\omega}$.

(6) For every free ultrafilter $\mathcal{U}$ on $\omega$ and every infinite set $A$, the ultrapower $A^{\omega}/\mathcal{U}$ has cardinality at least $2^{\aleph_{0}}$.