Firstly, it WILL get better!
I think you need to change your whole way around. It doesn't sound like you are getting huge support even though you have people around. Change this. Let your older child go to grandparents, even just one day or overnight, just to give you both a break. He will get one to one, you will relax a little.
Don't stress about cleaning/cooking. You will survive with a bit of dust and easy meals (cheese/beans on toast, jacket potatoes etc) Let all that go and concentrate on you and your children. If people are around, go grab a power nap! Your children will be fine with others and you need to recharge however you can
I picked up on you saying you were left to cry? Are you overcompensating? A little cry will not hurt the baby, walk away and make a cup of tea, talking/singing to him whilst you are doing so. He won't feel abandoned and you get 2 mins away
Also picked up on the fact that you said he needs total silence to sleep? Change that straight away. Keep the radio on at all times or constant CD's or even the TV. Low/medium, constant background noise. He will get used to it and once he is used to background noise, little noises won't keep disturbing his sleep.
If he has brief happy moments, use them to get him to lie in his cot or pram. Let him have happy times in there so that it isn't only a sad place to be for him
If he is happy during feeds, you can sit with older one and read a book, fuzzy felt or even play I spy - that way you aren't relying on the tv (although I did ALL the time - 4 in 5 years) Older one will still then be getting nice mummy time and you won't feel so guilty
He's 6 months, so I agree with the people above, finger foods in the high chair or some pureed food might sit in his tummy for longer. I know that it can also be less refluxy on solids rather than just milk
It isn't possible to be all things for all people - something has to give - it doesn't make you a bad mum. Make the people around take some more of the slack - you need to protect yourself here
It WILL get easier