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Without knowing more facts I cannot give you your chances, but I can tell you what the courts look for in gaining or providing custody. First of all, they DO NOT look at adultery or extra-marital affairs. While in the past that would be deemed a possible claim to providing evidence that the other party is morally unfit to raise the children, today that is no longer looked upon. What the courts look at, instead, is what they deem to be "in the best interest" of the children as well as the fitness of parents. Fitness is looked upon as possible history of abuse, neglect, domestic violence, drug use, alcohol abuse, mental illness, criminal record/history/parole/probation, & moral turpitude. The courts also tend to favor parents who wish to reside in the current location, and those that have a proper home for child-rearing.
For you to prevail you must show the other parent as unfit. Possibly using her late-night trips and lack of attention towards the kids as "neglect", and claim that since the children have a stronger bond with you, you should be granted a bigger role in the raising of the child. The home, on the other hand, is usually granted to the party in whose name the home is listed. If in both, you can request it, but she does not have to comply.