“How likely do fathers get full custody of their children?”

That is a very tough situation. Your question is difficult to answer because many factors are at play in family court. In Nevada, it is unlikely that one parent will receive full custody of their children. Even if there are allegations that one parent might be irresponsible, I have had cases where the allegations were founded, and still, joint custody was the deciding factor. I would say that it has been scarce where I have seen one parent receive full custody, and in those cases, there was a clear termination of parental rights on the part of the opposing parent. In Nevada, the courts prefer joint custody. If you feel that your wife is "irresponsible," you must prove that she is an unfit parent. If you have not already done so, the best recommendation is to speak with a family attorney specializing in divorce proceedings and child custody. You will most assuredly get a better sense of what your chances are.

Depending upon where you live you may want to engage an attorney to help you design the custody agreement you are seeking. It is not unlikely for fathers to gain full custody. It can happen. Most family courts offer a list of low cost or self help references. Look for a parenting support group where you can find others in your position who can offer advice from their experiences.

When considering custody for children, their safety and well being is top priority. I would consider with a lawyer with the evidence you have of things putting them at risk when they are in their mother's care. I would also do research on how your state views this concept with family law case results with similar circumstances and prepare your position with legal support.

Wondering about how often fathers get custody; it depends on the county you live in and what the trend is in your area. I know that is a very vague answer, but that is what I have seen.
Also, it's been my experience that you need solid evidence that your boys are not safe. A previous drug abuser is not relevant (at least in our county). Their mother being irresponsible is not enough evidence. Also, your lawyer is very significant. In my experience with the court system, the person with the best lawyer "wins." The court system can be really frustrating with lawyers and GALS not doing their job, long waiting periods, making decisions without enough information, and lawyers not getting back to you. The court system is an adversarial system there is a winner and a loser. So, I know this is not a solid answer, I would advise that if there is any way you, and the boys mother, can work things out together, without going through the courts you may be better off. Also, the decisions made are in place for at least 2 years and it is hard to change a court decision. It can be very costly.
Good luck.

Depending on the state in which you reside. some states are conservative on their approach to the question of where the children reside and typically with place the minors with the mother. if she has had issues in the past of neglect and has legal issues regarding this, it can be taken into consideration. I would bring any relevant information you think can support your case with you when you meet with your lawyer. Good luck.

This varies state to state and you really need to consult your attorney.