Let $1<p<\infty$, and suppose that $f_n$ converges to $0$ in the $L_q$ norm for all $q\neq p$ where $1\leq q<\infty$? My question is, is it possible for $f_n$ to not converge to $0$ in the $L_p$ norm?
If there doesn't exist such a sequence of functions, does there at least a sequence of functions which does not converges to $0$ in some $L_p$ for $1\leq p<\infty$ but converges to $0$ in every $L_q$ for $1<q<p$, and another sequence of functions which does not converge to $0$ in $L_p$ but converges to $0$ in every $L_q$ for $q>p$?
No such sequence exists by Hölder's inequality. This is in Folland's book.
For the second question, you can do something analogous to what I did in my previous answer.