Hi Raheel. As per my knowledge, it's very essential for us to know are you a shia or sunni, before giving any suggestion. Bcz, under mohammadan law, among shias free consent is required to make khula a valid one. But among hanafi muslims, khula taken under compulsion will also become valid. You told that your wife has taken your consent (Signature)) forcefully. But the law says, a muslim wife can take khula but she has to make the payment of some consideration to husband as she takes the initiative for dissolution of marriage. So have you given your consent after taking any consideration from your wife? Because any offer of khula must be accepted with the consideration (evaz) for the release.
But the reason which you have given here for the khula is not a satisfactory one in my opinion. If you really want to live with your wife and children, then i think, you can make some adjustments with your wife right?? Please think again about the conditions which you have imposed on your wife. At present everywhere we can witness people speaking about women empowerment, rights, freedoms, etc. In this kind of situation, how you can expect your wife to be within purdah all the time and to not to even see/speak with strangers?? Don't you trust your wife?? She is your better half. Plz try to respect her feelings and freedoms. Even among muslims, i have seen husbands respecting their wives and now a days people are coming out of such restrictions, customary compulsions. Hope you will not take me in a wrong way and try to understand your wife.
As per mohammadan law, among hanafi muslims, mother can have the custody of her children , in case of son, till his 7 years, in case of daughter, till she attains age of puberty i.e, 15 years. It varies in case of shias. Any how as per the age you have mentioned above, the children can be with their mother right now and you can't claim their custody as they are of tender age.
Hope this reply will help you. All the very best. I hope you will change your mind and get back your wife. Good luck