Question: What has ended up with Max? Billy promised to go back and release him, but we never see that happen, and no explanation is given for that in the Epilogue. Is Max a real figure? If he is, what has happened to him? Did he manage to escape, or did he die in prison?
Answer: It's never stated what happened to Max. The film was a heavily fictionalized version of Billy Hayes' book, and the Max character appears to be fictional as well or at least a composite of other real-life imprisoned Westerners that Hayes met while in a Turkish prison.
Question: When Billy is walking down the steps towards the door he escapes through, the prison guard throws him the keys for the door. What would Billy have done if the prison guard didn't throw him the keys? What was his plan, or did he even have one?
Chosen answer: Billy's plan was to bribe a guard to help him escape, using the money his girlfriend had smuggled to him. That plan failed, and he was almost raped by a sadistic guard. When Billy accidentally killed him, he used that as an unexpected opportunity to disguise himself in a guard uniform and walked out the front door. If he had been caught, he did not have an alternate plan and would have been severely punished, probably even executed.
Answer: Wasn't there a shelf near the door with keys on it?
Answer: He had a gun on him.
He also had most of the money in his shoe - he only offered one $100 bill to the guard and he hadn't undressed him yet.
Answer: In the book Billy escapes in a completely different way. If memory serves me he was on a work detail and escaped using heavy rain as a cover.
Actually, there was an episode of "Locked Up Abroad" where the real William Hayes told his story, and what he really did was swim off the island prison. He got to a boat, then snuck into the boat and sailed about 17 miles to shore. Then he took his shoes off to throw off any scent that the dogs would have picked up and made it across another river to Greece, and it was within a few hundred feet of the guards on the bridge separating the two countries.
Question: When they start exploring the catacombs under the prison, where did they get a flashlight?
Answer: I don't recall that it was explained, but it was probably stolen from a guard at some point and later traded from another prisoner.
I could assume that, sure, but I was looking for a more precise explanation. The more likely explanation is that they needed a light and the writers hoped no-one would notice. Seems like they could have used some sort of burning torches instead.
There is no precise explanation because it's not revealed in the plot how they got it. What they "could" have used is fan speculation, which opens dozens of possibilities, none of which are substantiated or relevant. They obviously obtained the flashlight, so that is what they used. How they got it, we don't know.
Answer: Exactly. How they got it, we don't know. Hence my question, to find out something I didn't know.
Answer: Max WAS a real character, and a Dutchman in real life, rather than an English one as portrayed in the movie. He eventually got paroled and later treatment for a severe drug addiction he had too.