If your income is "very Low", the court could determine that it is not appropriate to assess any of the charge for the guardian ad litem against you. Assessment of the cost is done at the conclusion of the case. You do not need to file anything in advance, but you do need to advise the court of your financial situation. As always, documentation is helpful.
An inheritance would not be "Shared" with your ex; however, the inheritance is income for child support purposes in the year it is received. If your ex learns of the inheritance, he/she can petition for an increase in child support.
The first hearing will be to determine whether the case is contested or not. If it is contested, a hearing will be set on a later date-several months in the future. The court will appoint a Guardian Ad Litem (lawyer for the child) and may order Home Studies of the parties' residences if living conditions are an issue. Psychological evaluations will not be ordered at the first hearing and in fact, are rarely ordered. If they are, the parties pay for them. The court also will divide the cost of...
If CPS has been unable or unwilling to take any action, then you are right to have filed a custody petition. You also are correct, however, that a judge is unlikely to move three children into your one bedroom apartment. This is the type of situation where a judge could look for other family members to care for the children, or even foster care, if you are unable to relocate and the father is found to be abusing the children.
Charges that are nolle prossed would not be considered by the court. If you know the facts surrounding the incidents, then there can be testimony about what happened. This could be relevant, depending on the facts.