You need to have a firm conversation with your DH, that if expects you to be a parental unit, you need to have each other's backs. He needs to understand that if you set a punishment, he sticks to it and if he sets a punishment, you stick to it. This is the only way the kids won't divide and conquer, pitting parents against each other. 

If he cannot accept that and keeps undermining your discipline, then I suggest he disciplines his daughter and you discipline your kids. And if he tries to undermine your discipline toward your kids or overly disciplines them without your permission, I would squash that immediately. No double standards can exist here. 

Are you sure you want to go through the adoption process of SD6 if DH doesn't see you as equals in parenting? It seems to me that he only wants you to be "Mom" when it suits him. 

A few things:

1.) Several folks will tell you not to adopt her because your DH isn't an effective parent and throws you under the bus. If hiding clothes and slamming doors are your biggest issues AND your DH is willing to actually give you full parental authority when you do adopt AND SD wants you to adopt her, then continue onward. HOWEVER, if you have any doubts about your marriage or your want to be her mother, not just a parent (because you can be a parent without being Mom), then take a pause on adoption. It's not a fault or weakness in your character to take a big, deep breath and evaluate the situation for what it is, not just what society says ideally it should be.

2.) Grounding doesn't seem to be effective. Time to come up with a different consequence, such as her losing access to her clothes and picking them out herself. If she can't take care of them, then she doesn't get to keep them in her room, and THAT means she doesn't get to choose her outfits.

3.) No 6 year old should be helping an adult with their disability to a capacity that a parent feels they need to compensate their child. To your DH, I'd say:

"DH, we all should help one another as members of the same family and household. However, helping one another shouldn't be rewarded with by decreasing consequences for a separate, unrelated action. Either SD is helping in a capacity that we all should be, which means there is no reward other than the natural consequence of a healthy, happy family, OR she is doing more than a 6 year old should and you and I need to look into finding you additional care at home. My job as your spouse is to help you in an adult capacity. That is not SD's job, and if you have relied too heavily on her to the point that you feel the need to compensate her, then you and I need to talk about how to better help you with your disability. It's not fair to put that on a 6 year old."

4.) An addendum to that conversation also needs to be about him undermining your authority:

"And DH, while we are talking about compensating SD in reduced punishment, I don't appreciate you making a unilateral decision to reduce it when I have instituted the punishment and I was the recipient of her disrespect. You trust me enough to adopt her and be her mother, so trust me enough to also deliver consequences appropriate to her age and the infraction. If you disagree, talk to me about it privately as I would with you. Do not usurp me with talking to me. Do not give me responsibility without authority."

5.) THAT last bit there is why I said you may want to better reflect on your relationship and your role as a parent versus Mom before adopting your SD. Unless your DH is willing to allow you to act as a mother, you'll be fighting an uphill battle that your SD will be able to manipulate and use to her advantage. It happens more often than folks would want to admit. If he won't let you treat SD as your own, then you have no business adopting her.

Even if you adopt SD, your DH is going to have to live with being a parent versus a dad to your son. There WILL be a responsibility and authority difference, and he needs to be 100% comfortable with that. I'm not convinced that he will be given the snippet you've provided, but you know your DH and marriage far better than I do. If you two haven't had a conversation about how your role will change with adopting SD but his role won't change in regards to your son, you need to. Your DH has already shown that he'll overstep your authority, so make sure he knows perfectly well what his place is in regards to your son and how it will be less than yours with SD post-adoption.

DD the way he does.  Not your way.  
First question is. Was your punishment of SD fair ? The amount of time of the grounding ?   Even if it wasn't , too long DH should of talked to you about lowing then time. Not go behind your back. And undermine your Athority.   
I would not adopt SD until this is worked out.  Or you could divorce DH and be on the hook for CS. For a kid you don't like. 
It seams like DH wants his needs taking care of.
The big Happy Family for his DD,  but not for your DS. Is he trying to be DS father.  Is he including DS in this Big Happy Family, Is he trying with DS. Does DH does one on one time with DS, ?  Sports, fishing, camping.  ?  Or only what SD wants.

OT - Try making her redo the laundry (with supervision of course). Much more effective in my opinion. 

Shaking my head you're making a mistake adopting her

Hmmm. The fact that he gives her special favors for helping him with his disability tells me that he may not allow you to be her mother in a real sense. It doesn't aound like a healthy dynamic. Maybe a mini-wife in the making. It's kind of messed up, seeing as how she is 6 years old. What does she help him with? Is it something you can't help him with? Does he truly need help with it? I agree that you should think long and hard about adopting her. If your DH is disabled and you divorce, you may end up having to pay CS. 

Hi! There were some good thoughts shared here already. One  thing I wanted to add was that it's normal to encounter these differences in parenting ideals. The best thing to do is talk about it, openly and often. This is not a one and done. I'm sure you've learned with kids that it's impossible to anticipate every curveball they can throw you so your going to need to keep adjusting as your family grows. No two partners are going to have the same viewpoint all the time, especially on a difficult  and complicated topic like parenting. Even with a bio kid that you share this kind of conflict can happen. 

Its hard to say from your post how bad the situation is. On one hand it may well be you are facing normal challenges that all imperfect partners face in raising children together, or it may be that there is more going on than I am understanding.

 I wish you well.