How many would say that their kids are out of control?

My kids are no angels and there have definitely been times that I have said "Things are out of control and I just can't do this" BUT there are also plenty of times when my kids show incredible amounts of responsibility and I know I'm doing something right. ALL siblings are going to fight... the extent of the fighting may vary but it's natural that they are going to fight. The key (for me at least) has been consistency in punishment and reward. I'm not always great at it... I'm a mom and I cave once in a while ...but I pay for it afterwards. I'm still learning.
My suggestion would be ... Lay down the law. Spell it out clearly for them. If you behave this way, this is going to be the consequence. Then carry through with it. Do the same thing for good behavior. It's going to be time consuming and frustrating but it does work.

I'd say Aaron is strong-willed but not out of control. He can be pretty stubborn at times but then again he's 4.

I do have my moments when I think the girls are in control rather than me, but I do my best to keep them under control. They may act up at home, but to act up outside of my house almost never happens. I'm gonna go through your post and offer a few suggestions, and I think Liz's suggestion of a parenting class is a good idea. I know that I have been to one, and read several books on handling spirited kids (My oldest daughter). okay, first problem... NEVER give into a tantrum. I know that is hard, but you have taught them that if they act out enough, you are going to give in eventually just to get them to stop. If you are out in public and they start throwing a fit to get something, tell them no... If they still won't stop, take them out of the situation. I have left a cart half full of groceries in the store (1x) and taken the girls home over a request for a certain food. If you are in a restaurant, get up and leave (stop and pay the check of course). DO NOT TAKE THEIR FOOD HOME... take yours to go if you have the chance so you can enjoy it, but let theirs get thrown out. Put them to bed as soon as you get home. Always give them one warning to stop the tantrum or you are leaving, and then follow through. It is hard, and since your kids are a bit older, it will probobaly take repeated incidents before the message sinks in and they realize it is not going to work... You will have to be strong.
You said that they are hitting each other, I am assuming this is a regular problem and not some rare occurence or you wouldn't be mentioning it. Something i have been doing with my two younger girls (much more likely to hit and argue than the older one because they are so close in age) is making them hug one another. If one comes and tells me that the other hit her, and they are acting ugly to one another I make them hug each other for a minimum of three minutes (the age of the younger one). They do it begrudgingly, and I have to remind them to hug not hurt... but by the end of the time they are usually giggling and friends again.
The swearing would be a major issue for me, if my children were swearing at anyone they would get hot sauce on their tongues. If the child likes spicy (Tessa does) find a bitter spice in your spice cupboard and put that on their tongue. That is my response to swearing and back talk. Kind of a different take on washing a mouth out with soap. I make them sit with it on their tongue for at least a minute before giving them any water to rinse the flavor away. I haven't had the swearing issue, but for back talk it worked fairly quickly.
If your oldest daughter is that defiant and has always been, I would take her to a child psychologist again. Maybe tape her acting out, since she seems able to trick people...

Basically you need to take back control of your house, and the key to that is the follow through. Make sure they know what isn't acceptable, and that the consequences happen every single time. If you stick to it eventually they will realize you mean what you say.
I know how hard it can be to feel like you have no control... we are here for you

also... a former member read your post and suggested that you look into Autism and aspergers syndrome for the oldest two. She has children with these issues, and it is close to textbook from what you posted. They don't respond normally, but many of them can do well enough at school to get it ignored.
She also suggested taping them, and showing them how they were behaving later, they may not realize how they are acting when in the middle of an episode.