Is a limit ordinal necessarily a cardinal?
No, $\omega + \omega$ is a limit ordinal. Its cardinality is $\omega$
More generally to Ross' canonical example, if $\alpha$ is any ordinal then $\alpha+\omega$ is the smallest limit ordinal which is strictly larger than $\alpha$. And if $\alpha$ is infinite then $|\alpha|=|\alpha+\omega|$, so $\alpha+\omega$ is not a cardinal.
Note that this is ordinal addition (as in Ross' example), and not cardinal addition.