I am sorry this has happened. Is he willing to also work on it? Did he stop his behavior and has he apologized? Is he willing to change?
When he does go out does he give you proof of all you request?
He should not be going out that much. It is not healthy for a married man to go out every night with friends while he has a wife and two small children at home. Of course, if he was cheating it would be worse, but regardless if he is cheating or not this is not healthy for him to be doing this. I would try discussing with him about why he feels the need to go out every night. Express how this makes you feel even if you do believe that he is cheating. From there you may get a better understanding if there is an underlying reason and possibly work at it. I would not take it personal as if he is unhappy with you. This is a problem within himself. He may not be ready to handle responsibility and he may not be mature enough as you had thought. You are right this is not a good impression for him to make on your parents. And if he has cheated on you in the past he should be working on that with you to prove he has changed as well as apologize. If he is in denial then it makes it harder to work through and get passed it.
That is terrible. It is not easy to deal with someone who feels they are dominant. In this case his entire thinking would need to change in order for his actions to change. You may be at the point to ask yourself is it better for me to stay in this for my life or leave now and go through a rocky phase, but be able to find future happiness.
Sometimes it is better for children to suffer the consequences of a divorce rather than having both parents and witness unhealthy behaviors and a distorted image of what a happy marriage looks like. This is something that you would need to analyze and be true to yourself deep within. Would he go to counseling? I am assuming he would not based on what you explained, but if he was that would be an option if you wanted to try and save the marriage. In anycase for the marriage to be saved he would need to listen to your needs and feelings as well as be sensitive to them and want to change. Then from there show that change. His actions tell a lot. Like you said one will show their love by their loyalty. That is true and along with other expressions of love and respect through action.