So, I have found that the getting homework done falls in to two really separate categories. You have to decide where on the range you are first then adjust accordingly. More on that in a second. First though, make homework important.
You have said:
We do not have a regular time slot for homework, because this would be difficult for a variety of reasons.
This means that your sending the message that homework is less important then what ever other scheduling conflicts are arising. This should change IMO. Homework and other educational concerns need to be the most important thing (or at least one of them).
When you were potty training, sleep training, trying to establish the "toddler routine" you made what ever you were working on the most important. "Go Pee pee" was something that was done very often. Special times were set aside for it. "Night Night" happened at a specific sequence of events. Homework (and many other things) are just like "go pee pee" and need the same level of importance.
Then we have the "what kind of parent do I look like if I don't..." thought. Screw them. You do what's best for you kids. Figure it out, what do you want to impart. Everyone else just needs to shut-up about it. Education is important, but you don't need to worry about what Mrs. Smith the math teacher thinks of you. You only need to worry about what your kids need, and what prospects a good education will open for them in the future.
You also asked, mentioned rewards. Back to "go pee pee" and "night night". Many parents give stars when potty training. well stickers are awesome to a 2 year old. A 7 year old isn't having that though. So up your rewards. If you get a month full of stickers you get the new video game. etc. You may need to keep it more immediate, you may be able to go longer, but rewards for good behavior have been a constant in their life, why should that change for this good behavior?
Now onto the getting them to do their homework.
Method 1 You WILL do you home work, and you WILL put effort into it, or else!!!
You need to be careful with this, but if the problem is just needing a nudge in the right direction, then this can help. If the problem is a learning issue, confidence issue, or something else, then this won't work well. But assuming this is normal defiance and you just need them to do their home work. It's time to put on the mean parent hat.
You will do your home work or you won't do another living thing! Nope, no talking. No games! No toys! You will come home, do your home work, eat, and sleep. Nothing else! And if I don't think you tried to do your work, I will make up new homework for you! Making a kid sit there and do nothing but homework, disallowing every single other task (except food, water, sleep, potty, and homework) will get them to do their homework. They may not do a great job at it. They may even fight you on it. But after a day or two of coming home, be sat down in the homework zone, and not being allowed to do anything else, they will do it just to get it out of the way. Then you can praise and give rewards. Don't forget the rewards that's important. If you have to go this harsh a route then make the rewards instant. Start stretching them out till you get to the "A's on your report card are worth money" stage.
Method 2 Fine what ever, it's your homework.
At that age, they are asserting independence, and this is one thing they have control over. So if your loosing the battle of the wills, or your not ready to fight it, then give up. Make a production of giving up, then tack on consequences. If you get a 0 in Math, I'm taking your X-box. If you fail a test, I'm taking you out of sports thing. If you get less then an A (adjust for ability) then were not going to Seaworld this summer.
You have to stick to your guns, but they also need to see that messing up in school has down sides. One or two semesters of sucky math scores is not going to be life altering. Not placing importance on education in some form will.
In both methods the keys are the same, bad things happen if you don't do good in school. Good things happen if you do good. The main difference is how far ahead you kids have to be able to "see" in order for it to click. Also keep in mind with method 2, that some kids just don't care if you take their X-Box. You might have to take many things before it clicks. Method 1 is more direct, but only really works if your trying to get over a hump. It's not a great long term solution, but it can help you get things moving.