Arhillis wrote: If she and Robert are at a complete stalemate on something, who gets the tie-breaking vote? Can Mary override a decision Cora has made regarding the staff?
No, Robert and Cora make decisions about staff as they are the master and mistress of Downton.
It's in the estate where Mary has authority as she owns half of it and works an an agent. Formerly Robert had the final vote but Mary and Tom could persuade him by making good suggestions. On the other hand Robert could make decisions without consulting Mary (Mr Drwe had to leave the farm and it was later given to Mr Mason).
After Robert's illness Mary became "Queen Mary" as Tom said. As seen in the episode where Downton was opened to the public, Robert could murmur but not change Mary's decision.