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If you do not mediate an agreement before the trial date, you will be expected to present your case to the judge.
Whoever filed the petition will go first. The petitioner will make an argument about what they want with regard to the split of assets and child custody. The way this is done is through testimony of the petitioner and anyone else that can support their position. Then the respondent will be allowed to cross examine the petitioner and the witnesses.
Then the respondent will do the same thing. If there are no lawyers, sometimes the judge will just take control and ask a few questions about what is at issue, and then allow testimony regarding those issues.
You don't need permission to mediate, but you can probably get a court order for mediation and continue the trial, if that is the route you want to go.
You probably do not need an attorney to help you determine what you are entitled to, but it may help you determine how to make it work. You are not going to be ordered to pay child support or alimony (given your respective incomes). The botXXXXX XXXXXne is that you should receive "an equitable distribution" of all marital assets. That usually equates to around 50% of everything accumulated during the marriage.
So if your wife is going to stay in the house, then you should get 50% of whatever equity is left after subtracting the 2nd mortgage. Same thing goes with retirement accounts. If she has a lot more than you in her 401k, you could be entitled to some of her 401k and IRA. It is not an exact science as there are ongoing debts and obligations as well to offset some of the assets, but you should end up somewhere in the 50% of everything range.
That said, an attorney may be able to help you if there are issues as to who owned what, and who paid for what. If you were married a long time, pretty much everything will go into the pot to be divided. But a lawyer may help you determine what was in the 401k or IRA's prior to marriage, how much the house is worth, etc. Having a lawyer will help you with the mechanics of your decree. For example, if you are awarded some amount of the home value... you need to think about how that will be executed. Will it be sold or will she refinance it and cash you out? Or maybe she will offset her claim to your 401k in exchange for the house...