Are device drivers state machines?
Many device drivers do read/write to memory.  For instance, consider a network device; it reads the packets to be sent from memory and writes the packets it receives to memory.  Or think of a hard disk driver; it does something analogous.   So, ultimately, it will depend on the driver.

That said, device drivers often do have a portion of their internal state that can be thought of a state machine.  For instance, the states could be something like UNINITIALIZED, ACTIVE, SUSPENDED, where it is initially in UNINITIALIZED state on boot; after it initializes the device registers, it could transition to ACTIVE state; on suspend, it might put the device into suspend mode and switch to the SUSPENDED state, and so on.  This only accounts for part of the behavior of the device driver, but it can still be a helpful way to organize the driver.  Or, in other words: often, some important aspects of the driver's behavior or state can be thought of as DFAs even though that's only approximately true.