BeginRead and BeginWrite are asynchronous. Once called, they send the request to the PLC, then immediately return to the caller. Once data is returned from the PLC, the DataReceived event is fired with the requested address and the data.
OwnerObjectID is a unique ID number for each driver instance. Since all instances share a common transport layer, the returned data also comes back with the OwnerObjectID, so each driver instance can identify whether it was data that it requested. In the constructor for the driver, it will increment the shared ObjectIDs and use the current value for its OwnerObjectID.
TransactionNumber and SequenceNumber are interchangeable terms. If you refer to Allen Bradley PCCC documentation, it refers to a TNS (Transaction Number). If you refer to CIP/EIP documentation, it will refer to SequenceNumber. Omron documentation will yet call it by another term.