OP I'm sorry I missed you and I hope (selfishly I grant you) that this reaches you.
It can be so hard with a partner who you love very much, but they have form for calling out your faults whilst refusing to see or address their own. Adding to that his recent habit of belittling your 'usefulness' at cleaning and tidying whilst you're pregnant and you're pretty much describing my situation when I was pregnant with my twins.
I had (and still have) four cats also two are old ladies who I've had since before I met DH - (he lost the 'D'H for a while during my pregnancy) one was his from before he met me, the fourth wasn't planned for but I found him as a kitten abandoned and took him in. At the same time I was struggling with long term ill health and depression. Things spiralled out of control at home and it became messier and messier, DH lifted a finger every blue moon but would berate me for the state of our home and continuously said "if you can't manage now you'll never manage when the twins arrive" - but did he help? Not much, not really, he required a pat on the head for washing some pots (whenever he deigned to do them).
Of course what he was saying about my messy habits was correct, I had gotten lazy with it, I felt overwhelmed to the point where I would see how much needed doing but just felt lost on where to start. Our oldest cat is a bit of a nervous one and wees when startled, so she would sometimes go before reaching the litter tray. Absolutely not the right environment to bring one child into - let alone two. But as you describe, I focused so much on how hypocritical my DH was being that I became childish and stubborn (not saying you're being either but I did) until one day I just decided "get on with it" - sounds so simple but that's what I did. Set myself a manageable list of tasks each day and did them. If DH felt it wasn't enough - Grey Rocked him and when he was calm - would say, well you know where the mop/bucket, hoover etc. is.
Then when the house was becoming more presentable I sat down with DH and told him straight - I will pick up my end - but he has to do the same if this was going to work. So we started off saying which jobs we loathed. For me it's washing pots, I have really bad atopical dermatitis and my hands get really bad when using any type of washing up liquid and I'm allergic to most types of plastic/rubber gloves. So he took that one, he hates hoovering - so I do that.
From there we split the tasks and made a checklist to keep on the fridge, eventually that turned into a rota, now we don't need it at all because it's become a natural part of our day to day.
As for the Dog, it does need walking more, but I'm not going to tell you to rehome as I think that with the right shift in your mental state and your partners, you can manage with dog and soon to arrive DC. If he would pull his weight, you wouldn't feel pressured into sacrificing the time with your Dog for cleaning, which you then feel resentful for because it (and I'm guessing here) is never enough in his eyes?
If he's not willing to do more, then stop doing his housework. Sounds petty but it might get through the fairness (or unfairness in this case). Such as, do your laundry, leave his. Wash your pots, leave his. Tidy up after yourself, not him. If he leaves unsightly messes in areas where you have guests, shunt them into his personal space. If he get's angry, then repeat "well if you tidied up after yourself it wouldn't be an issue!"
Then use whatever time you'd usually spend sorting his mess walking the dog! 
If he goes on a tirade, walk the dog.
Also, crate training is useful, my MIL works with training, walking, grooming dogs etc. she told me (when we had to rid our home of the cat wee smell from our eldest cat) that mixing one standard cup of wash powder (preferably bio washing powder) with half a cup of disinfectant with hot water would help rid the area of the smell - but you really have to scrub!
After you have removed the smell pads or plastic flooring in the outdoor hallway (if I've understood that correctly?) will help keep the area clear of smell if dog has an accident whilst retraining.
Positive reinforcement when she makes it outside is also a good motivator.
I was told by many people I would absolutely have to rehome all my cats when my twins came along, even the HV was certain I wouldn't manage. But I did, and still do, because I love my cats, they helped me through some really dark and horrible times. My eldest female cats were the reason I left my very abusive ex. So I get the reasons you want to hold your dog close.
Persevere, you can do this, tell your partner he needs to buck up pronto and remain firm. If he starts to pile on you again about what he deems useful then bite back. I wish you strength and luck.
Last tidbit of advice - though this isn't really healthy I'll admit, when DH and I were going through the early stages of this I would direct all my anger at his total hypocrisy into vigorous cleaning :')