First, I'm sorry to hear that you're in this situation. Sadly, you are in good company. Because it's not my place to offer legal advice, I will share with you my own experiences, and hope they help to point you in the right direction.
To answer the first part of your question: Judicial or nonjudicial? It's both. As I am no expert, I'm placing a link that will, hopefully, explain it in better detail than I. http://www.foreclosurelaw.org/Nevada_Foreclosure_L...
As for ways to stop foreclosure, it seems banks are becoming increasingly willing to work with homeowners. Communicate with them, honestly, about your situation. Have them explain what your options are. Some may include a reduced interest rate, a payment plan, etc. The options depend greatly on the banks' discretion. I have not yet spoken to a person who has successfully transacted a short sale--usually due to the bank's denial of the offer. Not to say that it can't happen. If all else fails, you may want to talk to a lawyer about bankruptcy. It's my understanding that you are able to keep your house, cars, and such, through bankruptcy. And, it puts an immediate halt on the foreclosure process.
Again, I'm not qualified to offer legal advice. However, I hope my experience is helpful in your decision-making process.
Hang in there!