Christ alive this sounds like utter shit.
No it isn't normal to cry every day at work and feel completely unsupported and undermined.
I don't know anyone who walks 5miles to work and back every day for the fitness boost 
I wouldn't, in the end, rule out moving somewhere cheaper so you can do something else - tbh this would be first on my list, not a last resort. Something has got to change.
I know that this isn't possible for the vast majority of people but if I were in your situation I would be on the phone to my mum and ask her to move to my town for a year month and be my childminder. No more getting any kids out of the door, just to do all that. There aren't many mums who either could or would though.
Then I know this is a real smug middle class thing to say but we nearly bankrupted ourselves paying a shared nanny for a year, and it was worth it 100%. Is it possible that it could be affordable given you are paying the childminder for 3 places?? For us, we wanted her 11am to 7pm and that worked brilliantly but what you want is someone in the house doing the children's routine. Then, bang, you are in work early, having had breakfast and feeling like your brain is in a single piece instead of shattered into a thousand ground glass bits.
I am surprised tbh that you didn't consider the antidepressants as a real possibility - you certainly sound depressed. Just because you have an obvious reason why you are depressed doesn't mean you aren't.
Someone who is sleeping for shit, eating for shit, earning all the money, with no support at work and home and doing motherhood to 3 chidren - this is not sustainable. Something will break and it will probably be you.