Tell him to get his own medical insurance and HIS kids are his responsbility, prepare to pull your weight...but your weight only financially.   Didn't you realize HIS kids do not have to do anything? Yes, they are entitled brats, this weakling parent has raised them to be irresponsible and lazy.  So clean nothing; cook nothing.  When the brats are around kind lady, nothing is about you with this kind of enmeshed sickness.  You do not exist and you are in the way of their love fest, I know this firsthead. Try your best to stay away from him whenever he feels the need to focus 110% of his attention on the nastiness he created.

So what does he expect your dog do? Get a job? Sounds like the dog has more utility than his kids---oh, I could have some fun with this one....

Do not let this man intimidate you into silence. You are describing a man so self absorbed (many of us know him), that he will stop at nothing to "make" you do --as he insists--around his brats, because he is wimp daddeee of the year.  Have no part of this, speak your mind, and let the cards fall where they may; if the idiot leaves and takes his brats--you can start over and have some peace in your life. You have to teach people how to treat you.....

He knows how bad the situation is with his kids (and him), so he's projecting all the badness on to you to make you feel that it's really your fault. 

I know you don't think of yourself as a control freak, but so what if you are?  Big freaking deal.  He's a lazy parent and a financial leech.  You're demanding about little things?  So?  You're also 100% responsible for that household right now.  You can be demanding fi you want to be.

What are you getting out of this relationship?  If I were you, I'd just be getting out...

OK I get how you are outgoing and him ... not so much. But consider dialing back the informstion you tell DH to pass onto his kids. Just tidy up after yourself and if DH wants to run around after them he can do it to his hearts content.

See the trick is to let the kids know or think that they are a non event in your life. It doesn't matter whether you are there or not. If DH is around he is in charge. When he isn't you are in charge. This is just the way things are. Otherwise he can pass the kids back to BM. Are his children left in the care of teachers at school? Does he expect them to follow the teachers instructions? What about babysitters? If a babysitter said "We need to get out now! There is a gas leak!" does he expect his kids to argue with the sitter?

As for the dog argument .... tell him your dog will get a job and his/her own health insurance although the dog my need assistance with transportation until he/she gets the hang of catching a bus or learns to drive. And my piece de resistance would be I know you think I am bossy. But right now I am supporting your kids and you. And I carry you and your kids on my health insurance. You should be kissing my butt because neither you or their mother are contribuing one cent. So back off on the criticism and man up on the helping out. I cannot live in ungrateful chaos. I can like in a happy grateful slightly untidy speace. When people are nice to me I can forgive or ignore a little mess. When people are hostile towaards me I notice everything."

He is unhappy with the balance of power in the home now that he is unemployed and probably hates the fact that he has to rely on you... but then again.. he is ok with soaking you  for money.. he just needs to knock you down a peg so that he doesn't feel like such a lousy husband (which is what he is.).

Now, he of course hits you with a bit of a truth bomb.... you are controlling.  I can see it through your posts and the way that the most minor things set you off.  Personally, I wouldn't let some of those things bother me the way you do... but we are different people and no two are alike.  But, as someone else said... so what?  A lot of people are more anal about the way their home is run and it's clear that "messy and ill behaved kids" are a huge stressor on you... so you have to set boundaries that work for you.  The bigger problem is that your DH doesn't agree with those boundaries.  He does a lot of  "yes dearing" when you tell him you don't want the kids sleeping on the couch.. or eating junk food but in reality, he cares not one lick about either of them and only is agreeing to basically get you off his back.  And I can see how your disengagement is pushing more responsibility on him and he doesn't like it.  Also, you would logically expect an unemployed guy sitting at home to actually pitch in to make up for the fact that they aren't working.. your DH.. I bet it's not happening unless you literally place the vacuum in his hand and turn it on.  But that reminds him he is unemployed and that makes him feel bad.. and he doesn't want to feel bad so he just retreats to his video games. 

I honestly think you would be happier without him.  You intensely dislike his children... and they are not going away.  they are going to be a thorn in your side as long as you are with him.  Your DH is also not respectful of you... I remember the car incident and others where he doesn't give a rip about what your needs are.  I honestly don't see any amount of tenderness towards each other.. in the way you describe him speaking and acting towards you and the way you speak of him.. it doesn't speak of love and comitment.. it's all conflict and upper hands and gaslighting.  My husband tells me literally every day how much he loves me and is thankful we are together.. how proud of me he is etc...   I know exactly where I stand in his priority chain.. and it's pretty much at the top of the list.  Not saying he has never done or said things to make me mad... or hasn't disney dadded a few times... but his feelings for me are never in question.  I don't get that feeling from your description of your life in your home.. it seems chaotic and conflict ridden... and I couldn't live that way.

He probably has a complex about you paying most of the bills. He knows he is inferior and now that you are telling him that he isn't up to your standards (fathering his children, participating in household mantenance) he is gas lighting you in return to doubt your own worth in comparison to him. Don't let him drag you down. Confront him. Sit down with him and tell him that you know things aren't going well for him business wise and it's bothering him BUT he doesn't get to take that out on you. He needs to recognize that when you are asking him for help it's not becuase you think you are superior than him but because you actually just need his help. And maybe you could reconsider how you ask him to do things. It seems like his male pride is a little dinged up, so you can throw in there some rephrasing. 

Been there.  DH thinks I am a clean freak, but I'm really not.  I just want people to take responsibility for their stuff; I'm not a maid or laundress.  I wrote a blog recently about this and I really, really cut back.  I don't say anything any longer and I really cut back on the cleaning up.  Yes, the living room gets gross.  But I just shove everything over to DH's area and let him deal with it.  His side of the closet starts falling into my way?  I don't pick it up, I shove it over.  I haven't totally stopped doing his laundry, but I don't retrieve clothes for him any longer - they aren't in the laundry basket, then they don't get washed.  It's been better for both of us.  In fact, I even left a hairball one of the cats threw up on the floor - he told me about it, I shrugged and said, "I'm going to bed."  (ew it was hard but I've picked up soooooo much disgusting old food from all over the house and kitchen floor I'm not doing it - he knows where the paper towels are).  Then in the morning, he stepped on it and ranted and raved...huh.  "DH, I'll clean up up, sure thing. It's gross stepping on things first thing in the morning isn't it?  See how I pick it up?  Easy."  I don't care if I was b!tchy one bit.  He got the point.

As for the expenses - I had the SDs on my insurance and now they aren't I warned DH that at open enrollment they were off my plan, I'm not the parent, I don't want to get reimbursed from him and BM - needing constant monthly reminders from me to pay me the h*ll back now.  If DH chooses to have less income for a time (he's employed via contract jobs and can pick and choose) then that's on him.  I pay for more, it's written on a separate ledger and he pays me back.  A couple hundred here and there is ok, but he got up to over $2K and that was too much for me...I need to pay off my debt too and that was not fair.  He just doesn't plan.

You need to lay off, and at the same time have a hard line, explained firmly.