You mention two very different problems.

1. Data race. It's a problem of mutation of some shared state. This problem is illuminated in the actor's approach by removing the shared state completely. Every actor has its own state and the only actor can change during handling incoming messages.

Even if we don't speak about actors and use message-passing within some other model (like CSP or Pub/Sub) then it hard to imagine how data race can happen in 1-to-1 interaction between entities in your application.

2. Deadlocks. There is no such thing as deadlock if actors use async message-passing. But there can be another problem: livelocks (https://en.wikibooks.org/wiki/Operating_System_Design/Concur...)

Or such thing as miss of a message. For example, actor A starts in state S1(A) and waits for message M1 to switch to state S2(A) where it will wait for M2. But actor B sends messages to A in reverse order: M2, then M1. If message M2 is not deferred by A, then M2 will be ignored by A in S1(A), then A switches to S2(A) and will wait for M2, but that message won't go to it.