Depending on the state you are in, it could be anywhere from two weeks after the court date, to two months. It takes roughly a week from the filing to the court date, as the tenent is to recieve summons no less than 7 days before the court date. The way that "pay and stay' works is thus-if the tenant has the $$ at the courthouse, the landlord must accept, and connot evict for any reason having to due with non-payment-however can still push for eviction based on other valid grounds. If they do not have the money at the date of court, it is then at the landlords and judges discretion on whether or not to allow the tennant a period of time-generally one or two weeks to come up with the money, or an acceptable payment plan, and if the money is ever not there when it is agreed upon by the tennant, landlord and judge, the cops can be called, and the tennant is OUT. The tennant can ask for a stay at court, but the judge will not grant an extention if the landlord is adamantly opposed. Also, if the tennant does not show, they automatically lose the case, and can also be forcibly removed by the police.
I would let the landlord know about the electric company notice, and let him know that they make you uncomfortable-the leaving of the kids alone might be illegal in your state depending on the childrens ages and the length of time-any illegal acts give the landlord more power over the eviction process and timeline.
hope this helps=also I am not a lawer, I just have a lot of friends and family who either own property or have been evicted/attemped to be evicted, so have done alot of research.