Partition an infinite regular language into 2 disjoint infinite regular languages
Every regular language is accepted by some minimal DFA. For an infinite regular language $L$, let's call such a DFA $M_L$. Consider any state $q$ which can be visited more than once while processing some string in $L$. If $q$ can be visited more than once, it follows that it can be visited any number of times. Define $$L_1 = \{w \in L \mid q\text{ is visited an odd number of times}\}$$ and $$L_0 = \{w \in L \mid q\text{ is visited an even number of times}\}$$
This is a DFA, so there's only one path. Any string in $L$ is accepted by the DFA, and must visit the state some number of times (maybe zero). The state can be visited an unlimited number of times; therefore, we know that there are infinitely many strings in $L_1$ (since there are words that cause the state to be visited 1 time, 3 times, etc.) and that there are infinitely many strings in $L_0$ (since there are words that cause the state to be visited 0 times, 2 times, etc.). Any given string is either in $L_1$ or $L_0$, and cannot be in both, so $L_0 \cap L_1 = \emptyset$. However, any word in $L$ is guaranteed to be in one of these two sets, so $L_0 \cup L_1 = L$.