So my thoughts are not likely going to fall in line with popular opinion. But I don't think this is such a cut and dry answer. Your feelings are 100% valid. But he isn't completely wrong either when he says his feelings count too. Both of you are having little needs neeglected, and your relationship is starting to struggle to continue to bond and grow. Especially with life and kids, its a easy pit to stumble into, but until you adress those little things, this situation can't be so easily fixed.
When he says he doesn't feel there is enough intamacy, that is likely true. It is also true it hurts so you hesitate. Neither of you are in the wrong here. Most people tend to agree, Intamacy is important. It comes with a lot of physical and mental heath advantages. That when it is there and then starts to disappear is greatly felt. Plus. Many just theorize it just helps with pair bonding, particularly if there have been other sexual partners in the past. However, your already trying to remedy your issues. So its okay that your hitting the breaks for now, as long as your finding other ways to bond with him. Focus and notice if his happiness seems to show less frequently. If he is too good at hiding his true feelings cleaverly keep verbal tabs. Even if touch isnt a mans main love language, most men still need it toa curtain degree to feel confident or connected. So thats his neglected issue, now for yours. He said your basically a corpse when he hugs you. Lol! I know that sounds hurtful too, but girl, we have ALL been there! Its even more common amongst men. Its because you have a long going on and are burned out. You give so much. Its like a glass of water: if you only have one glass of water to drink each day, you sacrifice drinking some moments to prioritize and drink during other moments you think need ot more. So when hehugs you, at that moment you just have nothing left you can give. Especially with nosleep, he isnt your priority. And that coldness he is talking about is him feeling it. Listen, your a mom. So obviously you can't always prioritize him. You got responsibilities and he does too. However! If he is noticing your growing cold and burned out, and wants tofeel higher on the priority list, he needs to talk and work with you to see where he can help ease your burden more. If your not sleeping enough, maybe he can offer to take care of the 3 kids and let you nap for a couple hours. Even if he only has enough strength to do that twice a week before he starts burning out, it makes a huge difference and willbetter allow you to spare more energy his way. I am sure he was a lot on his plate too, so going up to himand saying something like "you need to do chores more!" Wont likely help or go well. But figuring out his stresses and what could easier help relieve themand him figuring out your stresses and helping to relive them will both ultimately make things both easier. It will even have an extra added bonus of him fleeling more bonded to you. Honestly, intamcy helps a man mentally and feel wanted, and sharing DNA does create a bond, but for a man NOTHING bonds more closely and strongly, then going through and over coming a trial with someone. Theres a reason why service men are so close their entire life. If you both go through different struggles at the same time it will drive you apart, but if he throws himself into your struggles as well, and youhis, you will create a unbreakable bond and connection. Great example is loosing a baby. If a couple cries and holds each other as they go through the experience they become incredible. However if one or both close off and they try to get through the feelings alone, the marriages typically end in divorce or infidelity. So in your shoes, i would come up with a few different ideas to pitch and see if he is willing to try one to help stop the burn out, and in the meantime, i would activly push myself to try to show him more focus and emotion during hugs and kisses. Even if its just sparing him more thoughts in my brain during the day, would help train my body to be less cold during those moments when the do happen.
as for his issue, also talking to him and seeing if you could potentially add some thing to sexually get him where he needs to be without hurting you Should help. Doesn't even have to be permanent, but until your able to be more sexually active without pain. I went through something similar, when we realized a lot of lifes strains where amplified by my lack of desire, we worked on figuring out a way to balance it. We tried tools and I tried different techniques until we found a system that we both liked and worked for us. Of course traditional way is best, but when it wasn't an option, having a not as great but still nice alternative kept our relationship and bond strong until I was fixed up... so even if it was brought up amongst strong emotions, I would try to not to take it too personally either. His statement albeit hurtful, is still a cry for help more then anything.
Now the son issues. From the sounds of it, he is a teenager. Which, all teenagers have a hard time. It's a time of crazy hormonal fluctuations and changes. And unfortunately for this young fellow, he has instability and bad home experience on top of it. He probably really does feel that way and struggles to feel like your house is his home. He would feel like that anywhere weather you took him over seas or not. Our minds are our worst enemy. And his mind is likely trying to protect himself by convincing himself that he is more of a charity case in your home then a actual part of it. As the saying goes, we all have inner demons, and his are saying stuff that are quite loud. It isn't that he doesn't appreciate all you have done either. They are things he will never forget, it is just the demons right now are speaking louder then your actions. His dad likely knew that his demons had been bugging him with this, and thought giving him a key would help and speak volumes. He should have asked you first for sure. That was his mistake. But now the mistake is done, the worst thing you could do is flat out take it away. If any action was to say that your home isn't his home too, it would be that. However he clearly has also given you reasons to fear. Not 100% sure what your concerns are for him having a key, but I am also sure they are valid. Having some ground rules is a great idea. However, as difficult as it is, I wouldn't spring them on him just yet. I would give him an opportunity to prove he is trust worthy. Then, the first time he does something wrong, sit him down and kindly explain why what he did isn't appropriate and give him a few basic rule/guidelines. Show him that with the power that comes with the key, comes great responsibility, and that he should learn to desire and be proud of being able to uphold that responsibility by choosing to do the right things. Don't make the key feel like the freedom that comes with it comes with a heavy price (like requiring a clean room 24/7 in order to have it, or saying he can't have friends over if he wants it.) But show him it's a act of trust and love. Get DP on the same page so that both his and your feelings on the matter can be met. Crappy situation, but good opportunities here!
So are you over reacting or is he? I could argue both yes and both no. I dont think anything specific that you have mentioned here makes anyone a definitive bad person. I dont believe DP is a bad guy, just makes mistakes. I dont think his son is beyond hope, just needs to figure things out, and I dont think your a bad partner or mother figure. Just all of you have baggage and issies your carrying on your shoulders, that are difficult for the people around you who love you to figure out. And yes. I say love because if you didn't love himand his son, all of this wouldn't make you feel as bad as it does. And DP wouldn't feel so hurt as he does, and his son wouldn't be as troubled as he is. The only real obvious answer here is You are all just broken imperfect people trying to make things work. Which isn't a bad thing. Good luck mama! You all are in my thoughts, and I am wishing you the best in this situation. ❤️
Ps. Sorry for the long post, I am a stranger, so take everything said with a grain of salt. In the end, you are the one who best understands your situation and what the best answer for you to do is. No judgment here. Just hopeful thoughts, and potentially helpful suggestions.