Finding out if languages involving counting and modulo operations are regular
This is essentially @Yuval's hint, expressed in a slightly different way. You've said that you can prove
$$
L_u = \{0^n1^m\mid n, m \ge 0, n - m \equiv 0\pmod{3}\}
$$
is regular. If so, then its intersection with the regular language denoted by $00^*11^*$ will be regular. Show that this intersection is equal to
$$
L_r = \{0^n1^m\mid n, m &gt; 0, n - m \equiv 0\pmod{3}\}
$$
and you're done. Notice that a similar construction shows that
$$
\{0^n1^m\mid n &gt; k, m &gt; j, n - m \equiv 0\pmod{3}\}
$$
is also regular for any $k, j &gt; 0$.

The same idea can be used on your second question.