Question: What ever happened to the female villain in the end?
Answer: We don't see what happens to Angel Dust after Colossus carries her (and Negasonic Teenage Warhead) away from the collapsing Helicarrier. Both women were unconscious and we can only assume that Colossus put Angel Dust down somewhere before returning to Deadpool for the end scenes. Her fate beyond that - perhaps escape, incarceration or even recruitment - is unknown.
Question: How does Wade have a photo of Francis for his board, if he doesn't know how to find him?
Answer: Because it's much easier to find a photo of someone than it is to find the actual person. Law enforcement agencies very often have photographs of wanted criminals on file, even if they're still at large. In addition, it's a pretty common movie trope for a hitman to be given (or otherwise acquire) a picture of his target, then be told to find him/her (which may or may not be based in reality...more likely, it's just a film cheat to give the audience a visual reference and help them follow along).
Question: Was Essex Corp the company that cured Wade's cancer in the film and tried to weaponize him? The end credits scene from "X-Men: Apocalypse" (which is set in 1983) introduces us to Essex Corp, where we see a mysterious business man (who reminds me of The Recruiter in "Deadpool") take a vial of Weapon X's blood and place it with the others in his briefcase. "Deadpool" is set in 2016. By then, Essex Corp would be very functional with experimenting on mutants (as shown in "Deadpool"). So I'm just wondering if there's a connection there, esp with Essex Corp and Mister Sinister apparently playing a much larger role in future films to come. Also taking into account that Ajax (Francis) reveals to Wade that they aren't making him into a superhero, but a "super-slave." This, along with manipulating genes to make them superior, are aspects Mister Sinister is known to specialize in.
Chosen answer: Ryan Reynolds confirmed in the "Honest Trailer" for the film that the organization is in fact, Weapon X.
Question: When Deadpool dislocates both of Francis' arms near end of film how comes the villain seemingly cries out in pain? I thought he had made it very clear earlier in the film that he doesn't feel pain any more, of any kind.
Answer: He seems to yelp a little bit, but I just chalked it up to being a combination of him struggling with Deadpool and also being more exasperated than anything else. Plus sure, he may not feel the pain, but he'd still recognize that having his arms broken/dislocated is a very bad thing since it puts him at a disadvantage in the right. So he could just be reacting out of frustration. I'd probably yelp or scream too if I was in that situation even if I didn't feel pain.
Answer: No and I think that was the joke. He was angry at that whole system so taking out everyone who worked for Francis was probably just him warming up for Francis. What possibly proves this is when he finally gets to 'Agent Smith' he actually asks "where is your boss?" He doesn't say Francis to him.