In general, a character declares their intention to move before any actions are taken, and characters can take their actions as if they or their targets were where they were at any point during the turn. 12 seconds is a long time, so if someone is moving around a corner or into view it is reasonable that some number of bullets went towards them while they were in the open. For exceptions to this, see Taking One For the Team and Diving For Cover. Often it won't really matter, but characters with the lowest Initiative Score declare their movement first.
Now lets say a monster with big claws is 10 meters away from a dude with a gun. If the guy with the gun has the faster initiative he moves after the claw monster. Does this mean that the claw guy can declare he charges at the gun gun and is now adjacent to him, so if the gun guy tries to move away he's making a disengaging action?Disengaging: A character can move out of close combat just as they can move into close combat. However, turning your back on a lunatic with a meat cleaver is dangerous. To represent this, a character moving out of Adjacent range with an opponent is not only treated as being adjacent with that opponent for the rest of the round, they are no longer actively resisting, so the threshold to hit them is just zero (modified of course by circumstances such as visibility and dodging).
Taking One for the Team: If a character disengages from close combat, another character still engaged in close combat with the same opponent can choose to throw themselves into harm's way. This redirects a parting attack to the brave character playing bodyguard (or sacrificial lamb). The threshold to strike the new target is unchanged.
If that's how it works wouldn't it be to the advantage of the gun guy to have a lower initiative, thus he gets to move away from the claw guy and stay out of adjacent range?
The line "and characters can take their actions as if they or their targets were where they were at any point during the turn." makes it sound like the guy with the gun could just declare "well I am moving away as if you were at your starting point instead of in my face", but then that leads to a pretty powerful kiting effect for the gun guy.
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Also how does After Sundown resolve two dudes making attacks while charging past each other? Like two cavalrymen with lances riding past each other or two ninjas dashing at each other and clashing in front of the full moon and then landing past one another. How does that interact with the 'disengage' rule?
Like that
