Your STBX is reacting out of anger and saying things without thinking them through. In an ideal world those who cheat on their spouse would pay a heavy penalty for the transgression. In reality, we live in a no-fault world and it is extremely costly to pursue a divorce on grounds. IOW: is your X really going to spend thousands upon thousands of dollars and about that much time to make you suffer for the choices you have made? He may WANT to, but in reality, unless y'all are extremely wealthy, he won't be able to.
How long have you been married? Has he been a stay-at-home parent during the course of your marriage? If you have not been married for a considerable length of time and/or he has been without an income during the marriage (i.e. you have been supporting him solely) alimony is not an issue.
Child support is determined by custody and is a percentage of the non-custodial parent's income. Traditionally, the mother get custody and the father is the NCP, but that is changing. Are there reasons WHY he could prove you unfit as a parent? This is where having an affair is going to come into play. He can attack your character. He has cause to do that. Something to consider would be 50/50 joint custody with neither party paying child support to the other (unless one has an income considerably greater than the other). Both parents provide for the children when in their care and the children spend an equal (or nearly equal) amount of time with each parent.
You do not want to go to trial over custody. You do not want a judge deciding what is best for your family. You and your STBX need to work this out and come to an agreement because you are the ones who know, love, and are concerned for the well-being of your kids. WORK IT OUT!!!
Is the house worth fighting over? Is it REALLY? It is likely that, assuming your incomes are at a similar level, neither of you are going to be able to afford the house on your own. Is it worth all the $$$$ to the lawyers to fight for a house you will likely not be able to maintain after your household moves to only your income? Agree on this too.
You do not want to agree to anything that you cannot live with later. Even if it drags your divorce out (mine took three years) it is better to get it right than to just get it over with. I would recommend asking your STBX to go to counseling with you. Not marriage counseling with the idea of saving your marriage, but couples communication counseling so you and he can learn to work together even though you will not be married any longer. It will help the two of you work through the issues you have and get you started on the right page for your post-divorce relationship.
If you are in such a hurry to end your marriage in order to move on with a new relationship, maybe you SHOULD agree to whatever your STBX is proposing. Your children need to be your priority right now. If you have moved on to a new relationship before finishing your marriage you are NOT putting your children's needs first. The children's primary custodian needs to ALWAYS have their needs as the top priority EVEN IF IT MEANS SACRIFICING WHAT HE/SHE WANTS.
If your kids need you to be celibate and single while they adjust to the new reality of their lives, YOU NEED TO BE SINGLE AND CELIBATE. Entering into new relationships while you are still married is NOT what your children need you to be doing. They need you to be focused on them and their needs. Having affairs is selfish behavior. Selfish people make poor parents.
Sorry to be so harsh. But in just the few paragraphs you posted here there are many signs that point a negative light on you. If you present this same image to a judge, your STBX might likely get primary custody of your kids and 20-30% of your pay to support them. Is that really what you want? Would being a weekend visitor to your children be worth it to "get it over with"?
Just my interpretation...... Jackkey