Finite automata start state
It depends on the definition. Some definitions of a deterministic finite automaton (DFA) are a quadruplet $(Q, \delta, q_0, F)$ where $Q$ is the set of states and $q_0 \in Q$ is the initial state. Therefore, it must be defined.
However, the definition of non-deterministic finite automaton (NFA) is sometimes seen as a quadruplet $(Q, \Delta, I, F)$ where $Q$ is the set of states and $I\subset Q$ is the set of initial states. Here, you can chose $I = \emptyset$ and get an automaton without initial state.
Since a DFA can be seen as a special case of a NFA, it is not necessary for an automaton to have an initial state. In both cases (DFA and NFA), the language recognized by an automaton without initial state is $\emptyset$.