How can we get the Byzantine Empire to change its flag back to the eagle flag of the Roman Empire?
Flags didn't as much existed than banners, more or less informal. It doesn't mean that the Roman symbolism wasn't still pretty much present : Paleologoi's banner did included eagles, which were a main imperial symbol and borrowed as such by everyone in the regions, such as Serbians, Albanians, Turks, etc.
SPQR wasn't used anymore in any occasion ?
AFAIK, the only occurences of the original meaning were made in medieval Rome, especially as a symbol of populo's sovereignty, a political statement if you will.
That said, you had others uses of SPQR, but with different words (both in Latin and Italian) such as
Sanctus Petrus Quiescit Romae (St. Peter is buried in Rome),
Sancte Petre Quare Rides (St. Peter, why makes you laugh?),
Sapiens Populus Quaerit Romae (A wise people cherish Rome), etc. which may be genuine misunderstanding or (at least for a good part of it) attempts at puns and humour.
I have read that some remnants of Latin phrases survived into Middle Ages.
You mean like virtually all the litterary production up to the XIVth century?