Lots of wonderful suggestions on this thread!

If I were in your shoes, I would put a good deal of energy into planning and structuring your days. I know its hard to do that when you have kids clammoring at you, and limited energy. But whatever energy you put into planning and structuring will come back to you times 4. I swear -- time put in on the front of problems like this saves a heck of a lot of time on the other end, when you are not having to invest the emotional energy in reacting.
I agree with the poster who said that you need to address the problems one by one, in detail, by making proactive plans. So I think what I would do is make a long list of all the behavior problems (and be very specific) and then prioritize them in terms of how severely they affect your family. Then start working on a plan for the item at the top of the list. What you will find, is that solving the high priority problems will also solve a handful of the low priority problems.
Its very important to be specific about the problem behaviors. Don't write down things like "defiance" or "disrespect." Those behaviors are too vaugh to effectively address. Write down, "cursing," "tone of voice," or "breaking things." Kwim?
Work on problems ONE at a TIME, and set goals with your kid's participation. Make a plan to celebrate when you (as a family) acheive that goal. Part of your plan to address each behavior should be daily mini-talk sessions with each child, or as a family, to check in and see how it is going. Try to stimulate some sense of enthusiasm, or at least -- keep it positive.
Picking your battles is important, but don't confuse relaxing about certain issues with letting go of your structure and boundaries. Different sorts of kids need different levels of boundary setting. I get the feeling that your kids need pretty clear and well defined boundaries. My rule of thumb is that issues affecting primarily THIER bodies are issues to "let go." Ie. -- It is generally a mistake to be overly strict about toileting, eating, dressing, and sleep issues. Intentional destruction of property, directed profanity, and hitting are boundary issues that I would be pretty firm about. Does that make sense?
Finally, I would look hard at your routine and schedule and put some energy into structuring that. The idea of getting out and getting exersize in the early morning is really important, and I bet would make a world of difference. Fresh air, even when its unpleasant out, helps a lot too.