VLAN Inheritance?

Imagine the following scenario:

    Primary Switch
          |
    (VLAN 3 Port)
          |
   Unmanaged Switch
  (Dedicated to IPMI
      Interfaces)
          |
         / \
        /   \
Server 1    Server 2
 (IPMI)      (iDRAC)

Would the clients on the unmanaged switch automatically become part of VLAN 3 when trying to communicate the the rest of the network?

Yes. Assuming that the unmanaged switch is just a dumb L2 device, it will be on the same L2 domain as everything else on VLAN 3.

Yes, as long as you aren't tagging traffic through that port, it is just a dumb L2 switch.