I'm in two minds here whether to be sympathetic or not. Part of me thinks much of your current situation is your own doing. At the same time you do have a shitty situation which you need help if you ever want to escape the cycle.
Firstly you ARE entitled to benefits yet you have seemingly given up after your "adviser" has cut you off. Now if it were me and I felt the other person hadn't followed their rules, I'd be all over them. Especially if it meant the difference between buying food for my kids or not.
Also, I know there's a huge crackdown right now to get people applying for jobs. And I think the key here is the word "apply". From what you've said and others have experienced, it doesn't seem "looking" is enough and they want to see application/rejection letters, see you going to interviews. I've no direct experience here but on first glance, it doesn't seem unfair to ask someone to "apply" for work. Now it might well be that everyone knows you are doomed to fail but at least the boxes are ticked and you get your money yes?
I don't know what's wrong with your 14 yr old, does he have special needs which means he cannot get himself to school? If so then are you entitled to DLA carer allowance or whatever it's called? Otherwise he needs to understand the situation you are all in and start stepping up. At 14, I worked in my parents shop every single night til 10pm after my dad got us into massive debt and fucked off disappeared. It was hard but I learned to be responsible.
Mealwise, why are you cooking them seperate dinners??? They eat what you give them or they starve. It's simple. I have 3 kids and if I cooked only what they all liked, it would only be chocolate cheesecakes! My DD(14) doesn't like meat, she'd apparently like to be a vegetarian (except she doesn't like veg either). My son hates veg and my DSS hates vary along with the wind. It must also cost you more to buy stuff for so many different meals.
You blame tax credits/nursery for your current credit crisis but didn't tax credits pay you back eventually? If so why didn't you pay the money off your credit card? If they didn't why aren't you chasing them up?
Lastly, you really need to consider moving your youngest to the closest school. Yes, it's not as good but you have limited choices here and this one is hindering your chances at a job. If that's truly your priority then tough choices need to be made.