First let me apologize for not giving you enough details upon which to base a recommended course of action. Actually, the answer you gave, even without specifics, makes good sense: go after the most serious problems first and then move down the line, without dropping the ball on the first set of issues.
I think this is why we seemed so overwhelmed by this - trying to fix everything at once was just too daunting a task. But by breaking it into discrete elements, it becomes more manageable and hopefully increasing the chances of success.
Let me give you a brief overview of what we are dealing with here. The boys are 13 and 15. The 13 year old refuses direct requests to perform certain tasks that, to both his mother and me, are not unreasonable. He is extremely discourteous to his mother and contemptuous toward me. Last week for instance, he cursed his mother when she asked him to clean up his room. I was raised in a conservative Christian home where behavior like this was unthinkable. Punishment, both corporal and non-corporal, was primarily my father's job and I see it here as mine as well; and my wife agrees. When I started dating my wife and first met the boys I was shocked by what their mother (and especially their birth father) allowed them to get away with. Believe me, I prayed long and hard before entering into this marriage.
The older boy is running wild. He was picked and arrested by the police six months ago for drunk driving. Not only was he obviously under the legal minimum drinking age, he didn't even have a drivers license or learner's permit. For technical and legal reason which to this day remain unclear to me, the authorities decided not to pursue the matter and dismissed all charges with a warning Talk about fortunate consequences to bad behavior! He could have been killed. He could have killed others, including the other boys in the car. This is but a sample of some of the trouble he has gotten himself into.
Despite all this, I love these boys as my own. They are both bright and have a lot of potential. They are our sons. But it breaks my heart to see a train-wreck approaching without taking some drastic steps to get things back on track. And I think adding the possibility of spanking to the mix here is definitely indicated. Sort of a "better late then never" situation. To me this is a case for some serious "tough love".
You are correct that the problem has been the tolerance of bad behavior. I'm glad you are willing to take steps to correct the problem even though it is late in the game. It will be rough, but it is doable.
Your greatest problem is dependent on your sons' relationship to the birth father. Is he involved in their lives? Will he undermine you and your wife's efforts to bring discipline into your children's lives? Starting late in life is hard enough, but if you have someone who even unwittingly encourages the boys to misbehave the task becomes nearly impossible. As an example, children will play one parent off another and when one authority figure is out of the loop, it is tempting to "buy" the child's loyalty by claiming he wouldn't restrict the child in such a firm fashion. Soon the child wants to switch households to gain the perceived additional freedoms. I hope this is not your particular situation.
In regards to the fifteen-year-old, there is not much you can directly do about his drunkenness and drunk driving as too much time as elapsed. The best you can do at the moment is to severely restrict what parties he may attend and when he must be home when he does go out. Given the severity of the crime, I would put him on "probation" for a year. During this time he must prove himself to be reliable -- no drinking, no drugs, no sex, no driving without a license. I also would not allow him to get a learner's permit or a license until the probationary period is up. Yes, he won't be driving as soon as his friends, but that is the consequence of his breaking the law.
In regards to the thirteen-year-old, when a reasonable chore is assigned and not done, then privileges, such as going to a friend's house, playing video games, or watching TV, should be suspended until the chore is completed. Sit both of the boys down tonight and tell them that after careful consideration both you and their mom have decided that allowing them to screaming and cursing at reasonable requests is not good training for being an adult. Nor will they be allowed to treat their parents in a disrespectful manner. Any such behavior will subject to punishment.
When it happens, and it will happen again very shortly, ask your wife to let you know about a half-hour after you get home from work. (Give yourself time to unwind.) Select two punishments, one that involves some time, such as a grounding or a temporary removal of a privilege, and the other that is short but painful, such as a spanking. Allow the boy to select which one he will take. If he violates the terms of the grounding or loss of a privilege, then he immediately receives a spanking, but the grounding or loss of a privilege continues for the designated period. If either of them resorts to any violence, they should immediately be punished with a spanking.
Expect to be spanking fairly often in the short term as the boys push the limits to see if you really mean what you said. This will be followed by a lull as they regroup, which will then be followed by another round or two of pushing the limits. Eventually things will settle down, but you need to get yourself psychologically prepared for a battle.
My recommendation for the number of swats is usually ten at the start, though if you want to use the child's age, then this would not be so far different. Doubling this number shouldn't be necessary unless you find that the boys are uneffected by the spankings. And I would wait several months before making that decision. By spacing the blows a few seconds a part, you can get by with far fewer than you might have thought as each blow becomes more effective. See "A Series of Questions about Spanking" for more information. In addition, you will likely find at the very start that you many have to spank multiple times during the day. The smaller number will keep things from getting out of hand and if the boys don't learn their lesson quickly, then in a sense they are controling the number of swats that they get.
For your part, try very hard not to take their misbehavior as a personal insult to you. You need to act as a neutral judge in these matters. Try taking the attitude that if this is what they want by their misbehavior, then this is what you are obligated to deliver. You may rather not, you might find it inconvenient at times, you are likely to feel sorry for them, but in the end, this will mold them into responsible adults.
Having mom step out of the house would definitely improve her emotional well being, though I would leave the choice of leaving or not to her.
While I have spanked a few teenage boys on behalf of their single moms when they got way too big for mom to handle, I have not walked anyone through the process, other than by notes.
The foul language was edited out and I definitely forgive you of the unintended offense. Profanity is more than a problem of politeness. It is something that has gotten way out of control in our society. See the article "Careful What You Say" to see why such language is condemned by God.