No, you cannot pick and choose what past due bills you can include in the bankruptcy. Certain bills, such as regular monthly bills which you are current on, do not have to be listed. However, you can pick and choose certain debts if you file a Wisconsin Chapter 128 petition.
Not in the way you mean. By law, ALL debt must be scheduled in your bankruptcy, whether it is a debt to mom or the bank. Every single debt has to be listed. You are able to "reaffirm" certain debts if you wish, but doing so, esp. on unsecured debt, defeats the purpose of bankruptcy to begin with. Regular, recurring expenses, like the power bill, typically aren't considered "debt."
You must include everything you own and everyone to whom you owe money. That means all your assets and all your creditors must be listed. If you have been paying back some creditors and not others in the 90 days to 1 year before filing, the trustee could potentially get that money back because you're preferring some creditors over others. The idea is that all creditors in a chapter 7 should be treated equally.
However, after the chapter 7 is over, nothing is stopping you from making payments on any discharged debt. We typically see this with debts to family members. Although they are included in the bankruptcy, you can continue to pay them. Just watch out for how much you've paid them in the year before filing the bankruptcy as filing could lead to some unintended consequences.