Jen, a lot of it is you feeling guilty. I didn't make my husband's appointments until I had to be the one driving him. When he was still able to get himself places he made them. When I had to do it then it needed to be to my schedule since I would have to take off from work. I'm assuming that your husband is still able to drive since he goes to work so he should be making his own appointments.
Since he isn't feeling well and may be forgetful remind him once, at a time he can call and schedule or go, that he needs to do something. After that, it is up to him to remember. I have to make my current husband track his own appointments (he has traumatic brain injury and can't remember things). I remind him more often than I would only because he truly can't remember stuff and is never combative about it.
If his Mom feels like someone needs to be doing something to get her son the care he needs perhaps she can help. You can assign her the task of getting him in for his blood test. This may get that job accomplished without it being on your plate and/or make her realize what is going on so she can be more sympathetic to you. Some of it may be your own guilty feelings but she really may be critical and until she walks in your shoes won't let that go.
Guilt is a biggy and it can eat you up. You need to know that his illness is not your fault. His lack of self care is not your fault. My kids were able to place blame where it belonged and never faulted me.
My ex made bad choices. He had bladder cancer. After several surgeries to remove cancerous cells he was told he needed to have his bladder removed. I told him that he needed to do this and I would stick by him the whole way. My current husband told him that he needed to do it so he would be around for his kids. My ex didn't want to carry a bag around and with his multiple sclerosis they couldn't create an internal bladder. He waited a year and then decided he might go ahead and do it (this was after he was back in the hospital for them to remove masses from his bladder that prevented him from peeing). It was too late though, the cancer had spread.
Most likely, your older kids get that you have tried and couldn't get their dad to do what was needed. Having them help with more responsibilities will make that really sink in. I mean if they can't get him to do it and you can't maybe it is just his choice. You also need to realize that you cannot make a grown man do what you want. Therefor it is not your fault.
Your husband's behavior will likely keep him off the donor list. They are rather picky about who makes the list because it is so hard to get organs. They don't want to put them in someone who doesn't take care of them. My BIL is meticulous in caring for his new kidney. He was way down on the donor list when it was discovered a relative was a good match so he got his kidney. Otherwise, he would probably have died before making it high enough to be typed and possibly get one.
I'm concerned about him beating the dog. Yelling, while hurtful can be explained to the children. After all, he is very sick, scared, and angry. Beating on something defenseless is not appropriate. Remember, the kids don't get a choice where they are at. Your first responsibility is to them since they are dependent on you for their well being. Being in separate homes may help give both of you some perspective and let the kids feel safe. You aren't a bad person if you leave someone who is ill because they are abusive. I have never had anyone say that to me.