Oh you poor love, you are having a terrible time right now. It will get better.
I think you need to allow yourself to take charge, as it sounds like you are living moment by moment in the crisis or your poor health, the cat's poor health and your sleep deprivation.
You need to take charge and allow yourself to make tough decisions in the best long term interest of your health and your family. I would rehouse that cat. He is obviously not thriving right now. If it is behavioural, maybe he's stressed living with another cat and a toddler, or maybe he picks up on your illness. Don't feel guilty. Rehoming him will give him a chance to live somewhere he is the centre of attention.
Meanwhile, confine him to a small area without carpet. Do you have a utility area or section of the kitchen? Can you store his food high in a closed cupboard so he can't rip it to shreds? Once you have sorted out the cat, call in a professional carpet cleaning company if you can, or hire a rug doctor and use neat zoflora on the stains to kill the smell then steam clean over it to get rid of the marks. But rug doctors are exhausting work, so pay a cleaner to do this job for you as you are not well enough.
You need to prioritise yourself. I didn't do this when Dc were small and I was in a similar situation and it didn't get any better until I did. You are unwell and trying to run a home. You need to get well.
Go back to your GP about the constant diarrhoea. Get yourself some very slimline incontinence pads for when you are out and about. Try to limit journeys for now - get all shopping online, and time your eating for when you get home. Can you ask someone to care for your toddler for a day a week while you catch up on sleep?
All this will pass. My life was similar to yours (no sick cat but very sick DS2 who never slept and threw up everywhere - even the ceiling! with a very energetic DS 1, and I had gallbladder issues. I felt like the walking dead.
Life is so calm now and has been for years.