I’m not sure about filing for divorce from Bed & Board under the pretense of being accused of having an affair. There are several reasons that you can file for this but I do not know if the reasons you listed qualify…that would be up to a judge or the attorney you hire. As for the abandonment issue, I believe that is leaving without warning and making no finacial arrangements for your 1/2 of the marital debts.
I can answer this about the mortgage. If both your names are on the mortgage then you are equally responsible for the payment, regardless of who has been paying. The only way for him to be solely responsible would be for it to be refinanced in his name alone, which I don’t believe you could do if he moves out. Or the possibility of post separation support or alimony if he makes considerably more than you. That may cover the difference of what you can not afford on the mortgage.
The marital home would be considered marital property and you both would be entitled to 1/2 the equity in the home along with 1/2 the debt.