Maximal set with respect to the finite intersection property

$\newcommand{\cl}{\operatorname{cl}}$HINTS:

(a) Your suspicion that maximality of $\mathscr{D}$ is used for the forward implication is correct. Note that if $x\in\cl D$ for some set $D$, and $N$ is a nbhd of $x$, then $N\cap D\ne\varnothing$. Note also that if $\mathscr{D}$ is maximal, then for any finite subset $\mathscr{F}$ of $\mathscr{D}$ we must have $\bigcap\mathscr{F}\in\mathscr{D}$.

(b) You must again use maximality: show that if $A\supseteq D$ for some $D\in\mathscr{D}$, then $\mathscr{D}\cup\{A\}$ has the finite intersection property.

(c) This is false as stated. Let $\tau$ be the cofinite topology on $\Bbb N$, and let $\mathscr{T}$ be the family of non-empty open sets: $U\in\mathscr{T}$ if and only if $\Bbb N\setminus U$ is finite. Clearly $\mathscr{T}$ has the finite intersection property. Using Zorn’s lemma or one of its equivalents we can expand $\mathscr{T}$ to a family $\mathscr{D}\supseteq\mathscr{T}$ of subsets of $\Bbb N$ that is maximal with respect to having the finite intersection property. Every infinite subset of $\Bbb N$ is dense in the space, and every $D\in\mathscr{D}$ is infinite, so $\cl D=\Bbb N$ for every $D\in\mathscr{D}$, and therefore $$\bigcap_{D\in\mathscr{D}}\cl D=\Bbb N$$ contains more than one point. This is actually the part of the problem where you need the assumption that $X$ is Hausdorff: use that to show that if $x\ne y$, then $x$ has a nbhd $N$ such that $y\notin\cl N$.