Can someone please check whether my solutions are okay? I felt a little confused with some.
If Mr. Jones is happy, Mrs. Jones is not happy, and if Mr. Jones is not happy, Mrs. Jones is not happy.
$p$: Mr. Jones is happy.
$q$: Mrs. Jones is happy.
$(p\to \neg q)\wedge (\neg p \to \neg q)$
Either Sam will come to the party and Max will not, or Sam will not come to the party and Max will enjoy himself.
$p$: Sam will come to the party.
$q$: Max will come to the party.
$r$: Max will enjoy himself.
$(p \vee \neg q) \vee (\neg p \wedge r)$
(I do not know what it means by "and Max will not" so I assumed statement $q$.)
A sufficient condition for $x$ to be odd is that $x$ is prime.
$p$: $x$ is odd.
$q$: $x$ is prime.
$q\to p$
A necessary condition for a sequence $s$ to converge is that $s$ be bounded.
$p$: The sequence $s$ converges.
$q$: The sequence $s$ is bounded.
$p\to q$
A necessary and sufficient condition for the sheikh to be happy is that he has wine, women, and song.
$p$: The sheikh is happy.
$q$: The sheikh has wine, women, and song.
$p\leftrightarrow q$
Karpov will win the chess tournament unless Kasparov wins today.
$p$: Karpov will win the chess tournament.
$q$: Kasparov wins today.
$p\leftrightarrow \neg q$
(I do not understand the logic for "unless".)
Generally your statements are OK, there are just few errors.
$(p \land \lnot q) \lor (\lnot p \land r)$, and not $(p \lor \lnot q) \lor (\lnot p \land r)$ (I guess you just made a typo), where:
$p$: Sam will come to the party;
$q$: Max will come to the party;
$r$: Max will enjoy himself.
$p \leftrightarrow (q \land r \land s)$, where:
$p$: The sheikh is happy;
$q$: The sheikh has wine;
$r$: The sheikh has women;
$s$: The sheikh has song.
$\lnot q \to p$, or equivalently $p \lor q$, and not $p \leftrightarrow \lnot q$, where:
$p$: Karpov will win the chess tournament;
$q$: Kasparov wins today.
Someones prefer to translate "unless" by $\leftrightarrow$, but I prefer the translation given here.