FA for |w| divisible by 3 or number of zeros divisible by 2
Only six states are required. Indeed, you can accept the language of all words of even length using two states, and the language of all words whose number of 0s is a multiple of three using three states. Using the product construction, you obtain a DFA for your language using only six states.

We can easily show that six states are required using Myhill–Nerode theory, by considering the following six words:
$$
\epsilon,0,00,1,10,100
$$
Details left to reader.