Something to consider is grocery delivery. Honestly, it's inexpensive, and if it gets you something on the table vs stressing about meals, it's worth it. My DH used to use it all the time when he was a single dad, and we've used it since in times of illness or other events. You should see if a local store will deliver, or will do that thing where they gather it and you just show up in your car at an appointed time. You've also stressed before about your child's hydration and milestones. Please re-read those messages and take a step back. It's perhaps not ideal, but kids can live on chicken fingers and hot dogs so I'm sure there are lots of meal ideas you can work with. As for DH, if he's on your case about the budget and the meals, then have him do the grocery shopping and/or sit with you to discuss meal planning. Maybe he has no idea what a pepper costs. Something we do is stock up on sales. Pasta for $1? We'll take 4 and save every time we use one of those boxes.
When you say getting out to socialize, what does that mean? Go to a park? Do you try to get out once a day or once a week? If she's in daycare, I would not struggle with socializing any other day just for being social's sake. Take that off your Must Do list if you are stressed.
Regarding daycare, please talk to the provider. You want to be proactive instead of being dropped for being late. Is there a provision to pay more to have her there an hour later? Can someone else pick her up? Can you get a taxi if you know that DH is running late? If he comes to get you first, would it save time to go straight to the center? Etc. Will this be his new schedule forever? Can he do things like go in early to get out earlier?
When you say you struggle with cleaning, what do you need to do in a day? What's your daily list like? Can you move things around? Can you occasionally hire someone? Our cleaning lady comes 2x a month to give us a good overall clean.
If it's a matter of a mopped floor or a happy home, go for the happy home. Spend some time with DH watching a show or a movie or just talking. This AM I got up early with DH because he'll be in class tonight and we won't have any time. I also send him emails or he'll call at lunch if he wants to chat. It's more important to us that we do that than if I have the kitchen table clear (which I don't).
It sounds to me like transportation is the biggest issue and you should prioritize your health and daycare. Is there a driving school that would pick you up while she's in daycare, either from your home or from the daycare? Are there any taxis, mass transit, carpool, etc. options in your area you could use? http://www.chooseyourwaybellevue.org/bus
Another thought is if you have her in daycare for socialization and choice, and don't absolutely need her there every day, is there any chance you or DH could pick her up at lunchtime so you're not rushing at the end of the day? You get a few hours, but the provider isn't unhappy?
Take a breath and try to see everything as individual pieces and not one big hurdle.
ETA: I know he works out of the home and you are in the home and have daycare, but if you are struggling, then where can he help? Even if it's just one thing he takes off your plate, it may help. It's his house and his child, too. Please ignore the person who said there's no reason for your house to be a mess. If you are struggling, then you are struggling. I still think that transportation is the first thing you need to resolve because after that you'll be able to deal with daycare times, groceries, and most of all, your appointments. Also see if you can get any phone or skype appointments. Some providers will do that.