CHeydinhal's looks like some weird snake/serpent thing. ... what has an eagle got to do with Bruma?
That's a thorned stem (of a bramble or similar plant). That's why the Thornblade is one of the optional rewards for rescuing Farwil, leader of the Knights of the Thorn etc.
Eagles are mountain birds, Bruma county is mainly in the Jerall Mountains. Fits perfectly.
...the flag would be the Septim Family Crest.
Indeed, the dragon in a diamond motif is on the banners in the IC, but it's not clear which way round the relationship goes. Are the Septims using Cyrodiil's symbol, or is Cyrodiil using theirs? Or is that only a symbol for the Imperial City (home of the emperor)?