I totally get where you are coming from. I used to hide in my bedroom when the skids were here. I am quiet as are my children. DH's kids were loud and out of control when they were young.
On the plus side you are not dealing with a BM in the mix.
I would suggest that you have SS see a therapist to learn coping skills. He can't possible be OK with the passing of his Mother. I took my daughter to see a therapist when we got divorced and it really helped her which in turn helped me. I told her that this was her very own special person to talk to and that what ever she told the therapist the therapist could not tell anyone else. A few times we had family sessions. It took about a year of once a week sessions that were eventually cut down but it really, really helped.
Being mean to the pets is not OK. There needs to be a punishment for that behavior!
Is SS hyper all of the time or can he sit quietly in school? Maybe a trip to the doctors for a check up is in order and you can tell them what you are experiencing at home on a daily basis. Maybe he needs somewhere to let go of that energy. A sports team maybe. Lying almost always never works. No one likes a liar. I would have this discussion with him and make the consequences for his lying more severe than if he would have done something wrong and owned it.
Most six year old little boys are a ball of energy and loud. They are all rough and tumble. I have a grown son and four grandsons ages 7,5,3 and 2. Most people cannot tolerate the 5 year old. He is loud, he cries, he screams, he picks on his siblings, etc. He is good if it is only him and he is the center of attention. He is good for me because I don't take his nonsense. I have put him in time out on multiple occasions and I do not baby him like his Mother (my daughter) does and that is why it works. Don't give up on this little guy. He is so young and I am sure that he needs you. I understand your frustration and I too would need time away from him. Is there any after school programs that he could be in so at least you would have sanity at work? Make time for yourself and also get a sitter and have date nights with your husband. It just sounds like you are all under a lot of stress. Good luck!

even if you care for the child and circumstances, he is not your child. its the job of the father to parent the boy. resist the urge to step in as it wont be appreciated and you will be resented.

I am an introvert also, and I can't stand to spend every waking second around people. And if someone badgers me, creates drama, or tries to invade my boundaries, I get overwhelmed and it takes even more solitude time for me to recharge. It's really nothing personal.

Kids can be very annoying when they are bored. It sounds like the needs to be in aftercare after school so he can spend some time playing with his peers. In the summers, he needs to be in camp. Seems like you are overdosing on time with SS. A break would do you a world of good.

This is what you can do..... you disengaged and Dh is not liking it, there's no BM..

Simply wait till DH cooled down and then say to him, Hon if we want this to work we have to do unite, meaning you will give me full motherly rights over SS, I can give him things to do, I can be his mother and I can discipline him when I find it is needed and you will have to back me up in everything.

If you see SS is disrespecting me and saying things like you are not my mother, then you will have to tell him, no she's not your mother but she's my wife and you will obey her. If Dh does not understand your way of parenting he can calmly ask you behind closed doors and you can tell him why you did what you did.

It's small things like SS taking over conversations, you saying, SS calm down Dad and I are busy talking it's rude to interrupt.... it's small things like SS please come and help me with the dishes, clean your room, pick up your toys, SS it's rude saying that.... not major issues, that DH will have to handle.

If DH agrees to it all, then by all means stop being disengaged, the kid is 6, yes he's from a broken home, yes his mother passed, but there's another woman who's prepared to stand in and teach him some life skills...

I would bring in a rule, for an hour a day, SS will sit next to me reading, while I'm reading or quietly do my own thing, for an hour a day SS needs to be quiet and keep himself busy with homework on his own, reading, no electronic games nothing, I would even take this hour as a walk around the neighborhood not speaking..just an hour a day to calm him down..

12 years seems like the specific number until SS is 18 and can leave the house. Perhaps your DH sees a lot of himself in your needs and feels guilty. Instead of admitting that he doesn't like his son much, he chastises you for the same feelings he's having.

Don't feel bad about needing quiet time. My kids are high energy and LOVE to talk about their day. But everyone knows that if Mama doesn't get 45 minutes of quiet time when we get home, there's no dinner because I can't function after a long day at work. You want fed? Leave me the heck alone for a while.

Stay disengaged, adult step kids are awful too, I just wish I had not invested 11 1/2 years of my life in someone who throws me under a bus for his kids, it really does not get any better x

"I guess only a bio parent can be overwhelmed sometimes. Anyone else who says the same is being unreasonable and hates the kid."

THIS. This is the problem. I'd love to think I could be totally engaged with my SD but it would also mean being HUMAN and not always loving every second of her behavior, attitude and needs. Just like a normal parent! Except that as a normal parent, you kind of enjoy the kid a little more- because they're yours, and that's kind of magical. Maybe if you adopted your SS and were able to be a real mom to him someday you'd find that sweet spot where he melts your heart and makes it all worthwhile. I don't know if I'd expect that, though. It's hard on both of you because he lost his mom and you can't and shouldn't try to replace her. But you could be very special to him.

It sounds like your DH can't handle the truth about his kid. Newsflash- SS is annoying sometimes! Just like all kids. You need a break! Just like all parents. If DH could get a clue you'd have a much easier time stepping into this role fully. It's when they want us to do it all and pretend that it just becomes utterly unbearable.

Sounds like you both need counseling because there is a disconnect in communication. The emotions and hurt feelings and built up tension are interfering with the process and there is a break down because of it. Counseling can help guide the convo and the therapist can ask the right questions and do a great job of facilitating productive and effective conversations that need to be had. There needs to be a level of acceptance and compromise on both parts. Its hard for step parents to create a safe place for themselves uninterrupted by step family things and its hard for bio parents not to take that personal. You are entitled to some alone time to unwind, decompress, or for whatever you want it for, just because. Dh needs to respect that. DH is probably overwhelmed and stressed out because he doesn't know how to handle this situation and feels bad that his son lost his mother and really wants you to fill those shoes. Maybe you can tell him you will do your best within your comfort level to support him in raising his child but that he needs to respect and understand that you are not his mom and will never be. If he can't wrap that around his brain then you both have a long road ahead unfortunately.