Not an easy one to answer, this.
If you had asked me whether it was a good idea for you to leave the home before you actually left it, I think I would have said that the best course of action is to sit tight, and be as nice as pie to your wife and kids ( which is not easy, I know ) but you don't want to give your wife any ammunition which she can use to have you excluded. On general principles, you should be nice to your wife and kids anyway. If you want to take out your frustration on something or someone, go to the gym, find a punchbag and take your revenge on that.
If you are legally married you have a right to live in the marital home regardless of whose name this is house is in - but it is open to either spouse to apply for a Court order, known as an ' occupation order ' excluding the other. By the way, in the unlikely event that you are not the sole or joint owner of the house, you need to register matrimonial home
rights as a matter of urgency - if this applies to you, I will explain what this is and why you need to do it.
You may ask why I would have advised you to stay put. It is that you have at least one dependent child, maybe more, you don't say. Under these circumstances, there is a decent chance that your wife may be given the right to occupy the house until the children grow up. However, if it makes such an order - which means you have to leave - then the Court has to consider where you are going to live. If you secure accommodation for yourself - you have solved the problem for them and your wife might get a better deal, than she might otherwise get. But sleeping on a friend's couch is such insecure accommodation that it would not count against you.
Actually, I don't think your wife is being too clever wheeling out her new paramour too early ! It could certainly have a bearing on the financial settlement. If she cohabits with this man, the presumption is that he will contribute towards the living expenses, which reduces her needs, and therefore reduces your potential maintenance liabilities.
I hope that this will be helpful to some extent, but I have to make the rather obvious point, sir, that my name is Mike and not Gipsy Petulengro. Going back to live in the house when the new paramour is being introduced, seems to me a situation which has the potential for trouble, and I can't comment on the advisability of it. You will have to make that lonely decision by yourself.