The big question is, will Octavian still be Caesar's heir?
Octavian was not Caesar's principle heir until his final modification of his will.
Pompey and Cassius will be the great men of Rome at first. We can assume Cassius still goes East and gets slaughtered with his army in 53BC as it was his great dream to defeat Parthia.
Why? Crassus (not Cassius) was largely acting in response to the military acclaim of Pompey and Caesar. They heyday of Pompey's military career is long behind him and his political reputation took a massive hit during the triumvirate. And now Caesar is dead, and Gaul is soon to be, if not already is, in full scale revolt. Crassus is already the big winner here-he has the wealth, the influence, and the support of a critical mass of backbenchers in the Senate (Crassus was playing the long game politically). He's already effectively the most powerful man in Rome, and Pompey's political career is almost finished now that the Catonians definitely will not have a use for him with Caesar out of the picture. Maybe he can revive his career by being sent to Gaul, but Crassus could easily work to block that.
oman politics was a sewer and unstable so you probably still get two or three individuals fighting over mastery eventually.
No you don't. Republican politics will continue in its cut-throat and semi-unstable state for awhile, maybe things calm down for a bit, until the next crisis arises in the next generation. But until then there is no rule that the Roman Republic has to descend into civil war in the 50s and 40s BCE.
Hmm. Theres Likey to eventually be a dictator that comes to power and makes reforms of some sort, as otl.
And what is there to suggest that this will happen in the current generation? The closest thing I can think of would be what happened in 52 BCE with Pompey IOTL. Civil unrest leads to a temporary single man rule to right the ship and pass a few laws (like the OTL lex pompeia which instituted a 5 year interval between the end of a consulship or praetorship and the assumption of a pro-magistrate job in the provinces. This was designed to help reduce corruption, as without the prospect of an immediate financial reward, there would be far more risk in profligate spending and taking on massive debts in order to win elections).
The war in Gaul was already happening. Would Rome continue it without Caesar (and with his death it could be seen as a failure for Rome)?
There's Roman troops stationed in Gaul at the time, that are now in danger. The Romans aren't withdrawing right then and there. There's also way too much potential for glory for the enterprising Roman politician lucky enough to win command of the Gallic War.