Syria was on the state sponsors list when it was created in 1979. So, unless the al-Assads are prescient, probably not. More likely factors are the US support for Israel, Syrian support for Palestinians, particularly Hezbollah, and possibly leftovers of the cold war.
As for US attempts to depose him, as far as I know, those only started after his 2011 crackdown on Arab Spring protesters, which set of the civil war. Syrian behavior after hypothetical help from the US to crush the rebellion, that would probably depend on how long a leash the Russians have.