No he is still alive, and he is met again.
Because Syrio Forel is Jaqen H'ghar.
Let's look at the evidence:
Ser Meryn Trant has no real reason to kill Syrio Forel. Sure he helps Arya escape, but it's entirely feasible that Meryn bests him in combat (particularly since he has a wooden sword) and simply takes him prisoner. It's even possible that Syrio manages to knock Meryn unconscious or escape from him, and turn himself into the dungeons as a prisoner.
Jaqen H'ghar is from Braavos, as is Syrio Forel.
Jaqen H'ghar can also change his face, and therefore assume any identity he chooses. What's to say Syrio Forel didn't simply change his face and name in the dungeon? No one from the Night's Watch party knew who he was, and there was no explanation either. He was simply in the dungeons for some reason.
It would also explain why Jaqen H'ghar was confined to the cage with Rorge and Biter. He seems entirely sane and well balanced, yet he is locked up with two obvious psychopaths. It's because the jailers knew exactly how dangerous he was from when he was brought in, having killed 2 fully armed guards with a wooden sword, and assumingly having an epic battle against a man of the kingsguard and living to tell the tale. Even if he had changed his appearance whilst he was locked up, the jailers would likely only remember that he's an excellent swordsman, rather than remembering his face.
We also know Jaqen H'ghar is an excellent fighter, just like Syrio Forel is, because he assassinates 3 people in Harrenhal for Arya, and that requires all of the same skills that Syrio Forel teaches Arya (agility, speed etc.).
It would also explain how Jaqen H'ghar immediately knows who Arya is, because he's met her before and spent a significant amount of time training her, so he would immediately recognize her despite the fact she is dressed like a boy. And even if he didn't, he would have recognized Needle, it was a very distinctive sword she carried.
Also consider: until the wagon he is being kept in sets on fire, he shows absolutely zero interest in escaping. He can change his face anytime he wants, he only needed to convince one of the younger recruits of the night's watch to undo his shackles, and he could change his appearance and escape scot free. He stays because he wants to keep an eye on Arya and make sure she is safe.
This also makes sense as to why he doesn't just walk out of Harrenhal later on, he could just change his face and leave the employ of the Lannister army at any point. But he sticks around to make sure that Arya is kept safe. Having spent so much time training her, he has come to care for her and wants to look out for her, without making it too obvious that he is really Syrio Forel.
Of course, this is all speculation, but the evidence for this theory is quite compelling.