What would have been the consequences of a Carthaginian victory in the first Punic war?
Depends when they win. Early on? As in, preventing Rome from landing and wiping out Rome's "fleet" (if you could call it that at that point) and the Romans just saying "screw it let's stick to land warfare" and bowing out gracefully?
Or in the middle, where you would have Sicily divided in 3 (not sure how you get Rome to bow out though at that point)?
Or at the end, with perhaps a victory at Drepanum leading to the Romans losing yet another fleet and deciding they will never take Lilybaeum, signing a peace that allows Carthage to keep that and nothing more?
Each scenario has wildly different consequences.
Would Carthage come to dominate?
Short answer: No.
Long answer: Carthage had absolutely no interest in conquering large swathes of territory. They were interested in trade. Taking control of Sicily was only a goal because it would give them a complete monopoly on trade coming into the western mediterranean. Even then, its debatable how interested they were in actually ever taking over the entire island. They have no reason to take large swathes of territory.
Now Spain was an exception for 2 reasons. Reason 1, the loss of Sicily and the indemnity cut into Carthage's profit margin significantly. Spain had lucrative silver mines that were irresistable to try and go for. Secondly, it was mostly a Barca fiefdom, rather than a concentrated Carthaginian effort. Hamilcar wanted to use Spain as a base to boost his family's power and prestige.
Would Rome be able to rise again and eventually win over Carthage or would maybe some third power be dominant?
Possibly.
Or would the Mediterrenean be divided between two or more powers?
Most likely.
What other powers than Rome and Carthage would you put your odds on? Egypt? A greek power? Macedonia? Another Italian power? A Spanish power? The Gauls? Could any other power than Rome have conquered the entire area around the Mediterranean?
A multipolar Mediterranean. Makedonia continuing to increase its power over the Greek city states, Egypt and the Seleucids duking it out with the Seleucids probably getting the upper hand, a balkanized Anatolia with the Seleucids mostly holding the upper hand, Carthaginian western Mediterranean, Roman Italy....
It depends on how bad Rome loses. If it isn't much then they could possibly rise to challenge Carthage again but if it is like what Rome did to Carthage then they would be finished. If Rome falls then Carthage would more than likely be the unchallenged ruler of the Med for at least a while, much like Rome.
Carthage has a pretty crappy record militarily. Remember, this is the group that were getting their asses kicked by
Syracuse up until about 20-30 years before the First Punic War. Also, see above for why this was not in their interests.
Canaan, possibly southeastern Anatolia. Otherwise, no.
Errr....Cyprus, Krete, meddlings in Greece...
If Rome somehow had been crushed totally at an early stage, either by Carthage, the Gauls or some other power, could some other power have been able to conquer the whole arera around the Mediterranean, and if so, which power? Would Carthage have been the mosty likely candidate? What other powers than Rome and Carthage would you put your odds on? Egypt? A greek power? Macedonia? Another Italian power? A Spanish power? The Gauls? What is the most likely candidate apart from Rome and Carthage?
No other power is conquering the whole Mediterranean. The Romans were unique in their mentality towards warfare, and more importantly, they were unique in the sheer massive manpower they possessed.