Sadly I doubt a judge will laugh him out of court. It probably won't even make it to court unless one or both of you pushes it to court (if an arrangement cannot be made between yourselves and your solicitors or through mediation). My dad is a senior partner in a law firm and specialises in family law / divorce. I learnt a lot from working in his firm through my teens, and sadly through him representing me in both my divorces and several of my friends divorces and my new husbands divorce.
Unless a social services (CAFCASS) report can prove he or his new dp are a danger to the dc then he is likely to get 50/50 custody (if he wants 50/50 that is). I agree that the messages you have found do not paint him or his dp in a good light, but I doubt that this will be enough for a court to not grant him overnight access. Details of the ow will be irrelevant unless he moves her in, and even then probably not unless they got married.
My advice to you OP i'd to make sure you get in first with the divorce proceedings. It is always best to be the petitioner, rather than the respondent - gives you more power and control. Chances are if your ex has seen a solicitor they will tell him to divorce you sooner rather than later to get a settlement sorted asap. As a rule, men tend to lose more in divorce (whether they deserve it or not). Its in their best interests to get financial and childcare settlements agreed quick - and the only way to get that done is to divorce.
R.e custody, there is no such term now. It is known as 'residency'. It is agreed who will be the resident parent (the childs first home / where they are registered etc) - which looks like it will be you. And who is the non-resident parent (your ex). Once that is agreed then it is about agreeing (either between you, or through mediation / the courts) what the access schedule of the non resident parent will be.
Once this is agreed it is in both of your interests to have this written up in a child arrangements order - which is a legally binding doccument that means neither of you can be messed about with access without their being some kind of penalty (fine, criminal proceedings). Personally me and my exdh had a very amicable divorce and sorted it all out ourselves with no mediation or court orders and it has worked well (divorced 4 years). However my new dh has been through hell and back with his exw..... thats another story, but you name it, we've been there!
Regarding finances, all your ex is legally entitled to pay you is the CMS calculations figure based on his salary - as long as he is paying at least this amount then a court will not be interested in how much he says he has or doesn't have, or how much poverty he pleads. He has to pay at least that amount. end off. Make sure you sort out all the stuff r.e house move / equity etc asap. This makes me nervous and which is why pushing through a divorce may be the only way to make things legally biding at this point in time.
legally whilst you are still married he has a responsibility for your joint mortgage. chances are he will have to stay on your mortgage if you can't afford to take it on by yourself. It will more than likely be written into your financial consent order that the house has to be sold / you own him his share (whatever is agreed) - when the youngest child reaches 18, or before that if you re-marry or co-habit with another man. The reason for this is that nowadays the courts like to make things separate as soon as possible - this is called a 'clean break' order which is what the family courts push for now.
This will enable your exdp to get his hands on his money quicker and afford his own place so he is not financially tied to you. What it can mean is that you can effectively feel penalisd for having a new relationship (moving a man in / re-marrying) - but then I guess if that happens then you'd rather own a house with your new dp anyway!
Good luck OP. You sound like a strong lady - you will need to be over the next few years. It will be tiring and emotional and it will change you. But get some good legal representation and be pro-active. But please bare in mind that there will be things that will be out of your control, the law is a funny thing, and you cannot control what your now exdh does and thinks etc. One of the greatest lessons I have learn through my experiences is to let go - which for a control freak like me has been hard! :) You lose a lot of 'control' when you separate / divorce. So hang in there and look after yourself x