I've slept on your problems Milamber. It is indeed a sad and tragic tale.
Solicitor
It's quite clear to me that you need a really good family law solicitor. You need to have proper concrete advice on your options. You're not going to get that on AAM.
Marriage of 3
From what you've posted it looks like you're in a marriage of 3, and that will not work. Even they way you speak on here about your wife and her father, you see them as a team, and you put yourself as them against you. Looks to me your wife has not grown up. Basically been fully supported by her father and then has never stood on her own two feet and married you and expected the same from you. As SarahMC so eloquently put it, it's amazing how a grown woman has been so infantalised by two adult males.
I'm sure this way of being brought up has had negative results for your wife as she strives to be an adult when she has never actually let be one. It must be very difficult to be so cosseted from the real world to suddently be brought at a sharp focus into it. You have to be understanding of that. And it may be part of the reason that you two are having such marital difficulties. She must be struggling with all of this too.
Wife's letter to you
This you need to keep in a very safe place, this could be key to preventing you losing your home or any other type of bully boy tactics such as changing the locks on your house. No way FIL or your wife will want this dirty linen aired in court. That's your ace. But your solicitor will have to tell you if it's good enough. I agree with your opinion that for young children your wife will probably get custory and 'possession' of the house, but it's not always so clear cut. Possession does not give ownership however. In a custody dispute, depending on the ages of the children, a judge will take into account their wishes along with their best interests.
Also it is not necessarily true that when your kids are 24 that your assets would be split 50/50. Judges nowadays expect women to support themselves too. Another posted mentioned bringing up kids and minding a home as giving some kind of ownership. It's quite the opposite, both those tasks have zero monetary value contrary to the 'special place' of women in the constitution. And as they have zero monetary value, then they do not give a spouse a right to equity in the home. Other things can and do give rise to such equity.
Divorce/Mediation
Do you actually want to stay married to your wife? Only you can answer that. Sounds irreparable to me, but I'm not a marriage counsellor. Hanging in their for the kids, the house where you cannot even speak is not worth it long term. I don't expect you to answer that on here, but it's something you need to think about.
Revenue
My advice here is as another poster mentioned and go the separate tax treatment route. Based on all you've posted I would forthwith cease any payment in relation to the apartment. It is not your problem and neither your FIL or wife seem intent on sorting it out in a mutually beneficial manner. As a husband you are providing a home for everybody down to spending every last penny on said family. More than this no man should be expected. Particulary where one party is not even engaging in a mess of their own making and blaming the other party for not sorting it out for them. That's just pure nonsense. You're wife I'm sorry to say needs to grow up. Normally I wouldn't be so strident on this in relation to a marriage, but what you've posted doesn't sound like a marriage of equals to me.
Andrew at legal in AIB - hope one of you geniuses are reading this.
What nonsense you guys in all banks are spouting at people. Again I'll say it, it's underhand of banks, it's deceitful and it's downright abusive to ordinary people who are struggling to tell them outright lies. How can you guys work like this and sleep at night.
OP is not liable in any way shape of form for his wife's mortgage and he cannot be pursued for same. To any others in the same position, believe nothing of what the banks 'say' to you, if they are not preparted to put it in writing, then you know they are lying.
Reporting FIL to revenue for tax dodges/issues
That only brings you down to his level, no need to do this. In any case it only risks making the situation worse. But your knowledge of it may help in FIL and your wife in not going to law with you.
I hope all of this has been of some help to you but the main advice is to get a good family solicitor so that you really know your options.