Do you think its likely that even before the doom, under pax valyria, the free cities still competed violently with each other using deniable assets like pirates or even neighboring states as proxies for their trade wars?

Violently? Unlikely. The Free Cities were described as “dutiful daughters, neither making war upon Valyria nor defying the dragon lords in any matter of significance; they remained willing allies and trading partners of their mothers.” I would say a trade war, even conducted through proxies, would have been seen as rocking the boat too much, risking intervention by the motherland.

More peaceful competition? Almost certainly.