Definition of topological space & open sets

No. Consider the usual topology on $X=\mathbb R$ with $A=(-\infty,0)$ and $B=[0,\infty)$.

$A\cap B=\emptyset\in\mathcal T$, $ A\cup B=X\in\mathcal T$, $A\in\mathcal T$, and $B\not\in\mathcal T$.

No. It does not imply that the set is open. For instance:

Let $X= \left\lbrace a, b, c\right\rbrace$ and consider the topological space $(X, \tau)$ where $\tau=\left\lbrace\varnothing, X, \left\lbrace a \right\rbrace\right\rbrace$.

Let $A=\left\lbrace a \right\rbrace$, let $B=\left\lbrace b, c \right\rbrace$.

Then,

$A \cup B= X \ \in \ \tau $,

$A \cap B= \varnothing \ \in \ \tau$.

However $B \ \notin \ \tau$.