I am sorry to hear of your loss. If your father truly left no assets (the house is not an asset, since it is upside down), a probate is not required in Nevada. Unless one dies with at least $20,000 of assets, the legal process is not mandated by Nevada law. The house avoids probate since one of the joint tenants survives. To clear the title, you simply must record with the county an affidavit of death of joint tenant, accompanied by a death certificate. That places the title in your name, only. Recognize that the bank may seek to exercise its "due on sale" provision in the mortgage when the title transfer appears in the public record, but so long as you continue the payments, that is unlikely.

Life insurance proceeds are also non-probatable as an asset of the estate. Those funds are yours, not your father's, and are not subject to the debts he may have left unless your name also appears on those obligations. Make the claim on the policy, get the death benefit proceeds and use them as needed to keep current on the bills. If you choose to keep the house, there should be no problem doing that if you stay current on payments, including insurance and property taxes. When the market improves, you can hope to refinance the property.

It sounds like your dad tried to get things straightened out before he passed away, and you should be grateful to him for making the effort to have it all set up for you. A trust or other estate planning instrument might have served him better, but it appears he did what he could under the circumstances and you should be able to go on with your life in the family's Carson City home.