Is the PLC register also changing based on something in the PLC or inputs/outputs etc., or is it just supposed to hold the value you write into it (like e.g. a setpoint)?

An explicit Write can be made at the same the data are being collected by the server, or in transfer to the client, or inside QuickOPC, or passed to your app. The stream of data that are flowing from the server to the QuickOPC client for subscriptions that are already set up is independent from any explicit OPC operations being called at the same time.

(for record, though not what the question was about, below is a copy of explanation of how it works with explicit Reads and Writes - when no subscriptions are involved) :

With the default settings, on the OPC client-server level, all operation that are requested by the client (QuickOPC) on a specific target OPC server will be serialized. The operations include explicit Reads, Writes, requests to Subscribe and Unsubscribe etc. So if you are referring to explicit Reads and Write, they will be serialized per target OPC server, even if you, in your code, use multiple threads to call them.