To start with you are working from home, you do not have the time to babysit his kids. Why doe he imagine you do? That's just plain insulting to you and shows his deep ignorance and lack of respect of what you do. Time to tell him that the kids are there to see him, not to be babysat by you and he needs to find another solution to take care of them because you won't. If he says he can't, well then, that makes it easier - he can take them to work with him. What? He can't? But isn't that what he's expecting you to do all summer? Tuff on him. THey are no longer your problem. 

If the kid continues to do what he does after being told not to numerous times and being punished, the punishment isn't working. Maybe electronics or going to the arcade aren't his currency. Has your DH tried other punishments, or more severe punishments? What he's doing now isn't cutting it.

Are there camps where you live that DH could send his kid to? You shouldn't have to worry about him when you're working, and he doesn't sound mature enough to be left to his own devices.  Where is his mother in this picture?

Why is SS in your car? I hope you aren't chauffeuring him around anywhere. 

SD used to make a mess all over the house. Basement, living room, etc. DH finally told her she can't hang our or eat in those places if she can't keep it clean. Well, of course SD is too lazy to follow her dad's directions. So now all she does is stay locked in her room. It's totally disgusting, but atleast it's in a contained part of the house. I don't go in there. 

P.S. - I would have flipped over the Sprite. too. He can't seem to handle a soda, so he had no business drinking one.

This kind of stuff can be relationship ending. 

Ask yourself: Can you continue to live like this? At your ss age, it is not likely he will change - even though your husband is kindof trying (he should have done more when ss was younger). What is your limit going to be? You have to really impress on your husband that these behaviors are not something you can live with, perhaps then he will REALLY step up. If not...again, is this something you can live with for at least 6 more years? 

Me? I'm not going to be worried about poop all over the house (or in my food - I'm sure he digs those nasty poop hands in chips, bread bag, on the milk container, etc.). Hell no. 

If DH is tired of hearing you complain then he's got to step up his parenting game. If current punishments don't work, try something else. Maybe instead of (or in addition to) taking things away the skid has to do something extra--anything from cleaning the bathroom to washing down walls to weeding to writing sentences or researching and writing a paper on nose hygiene. And your DH needs to be the one to supervise.

I can see how disengaging completely will leave you living in a filth pit. But backing off where you can and turning it over to DH is a reasonable approach. Can you go visit friends or family for an extended period so DH has to deal with him 24/7?

Do you think he's just extra awkward and immature, or are there health/mental health issues that need to be evaluated?

My OSD was very messy, left trails behind her and all. She didn't care about anything around her. And YSD with bathroom issues. SIGH. Yes, I successfully disengaged so DH would handle it, not me, and gave both of us I think some relief. I can offer you some examples of what I did.

1. Never drive a skid anywhere if they cannot keep your car clean. The last straw for me was OSD putting a cup of soup (oatmeal or ceral - whatever) on the floor of the moving car and of course it tips over when I brake. No more skids in car with anything but water. I opted out of driving for the most part later but that's another issue.

2. Kitchen - if skid cannot clean up or your DH thinks the poor baby just cannot do it for whatever lame reason, then stop. You have choice one which is to not touch a thng unless your DH or SS cleans - you may not be able to make meals so be it. Or you have choice two which is to clean up, but put all kitchen items you had to clean yourself into a box or something and it goes away - because no one cares enough neither do you - you may not be able to make meals so be it. I donated a lot of stuff to our local charity shop this way; DH had to buy more stuff.  The good things I just kept put away so NO ONE could use them.

3. Living area - Anythng left out for more than 24 hours and it's obvious no one cares. You get to decide what to do with it (trash, donate, recycle or keep in a special place to see if anyone noticies). Again, I donated a lot to the local charity shop. Not ONCE was anything missed.

4. Personal skid room - don't bother with it unless it starts smelling or there's a physical issue like mold, rotting food. and vermin attracting things - this all will smell first from the hallway. Give your DH and SS one warning, if not cleaned up by X date because you will not live in a house that's treated as a trash heap nor will you be subjected to vermin, then you will clean up as you see fit and they have to live with it. I trashed bagged OSD's room 2 or 3 times - DH and OSD hated it but after the first time in which she tried saving items one at a time from the bags in the garage rather than putting them away properly as instructed, I they disappeared. I lost all caring in this matter due to the level of filth. There is a dfference between messy (ok) and filth (not ok).

5. Bathroom - if it's a shared bath then YES the adults need ot check on it each and every time he uses it just like he's 4. Because that's the age he's acting. Recently had a pee issue with YSD and the only way this was solved is that DH stepped up and laid down the law and checking EACH time she used the bathroom. And SHE had to clean up. I told DH straight up if he did not handle it then I most certainly would because human waste is NOT acceptable in any way to be spread around.

6. No more going out to eat with skids until they learn how to act like decent humans in public. End of story. I did this successfully for almost 2 years.

You can disengage from most things, it's hard at first but gets easier. Think it through and always have a valid reason, a clear explanation and expectation.  Boundaries are your firend.

I wouldn't know how to disengage either with this because it affects your whole house. My sd12 is nasty. She has to do her own laundry but she will lay in period stained sheets, eat cereal in her room and leave milk for weeks because I quit saying something dh. But when it affects outside their rooms or gameroom then it affects me. 
 

The only option I see is dh has to find childcare since you're working too. He doesn't stop to clean up after filthy people while he's at work and you shouldn't either. I'm so sorry you're in this position. 

This is disgusting and intolerable. I couldn't live like this at all.

It's his kid and he is doing nothing.  So what do you expect ?  First. DH must find childcare for his DS for the daytime, when he's is working.  Sports camp. Regular camp.  
Second you must find out what really matters to SS. His phone, tablet, gameing system. ?  Something you must figure out.  Then he loses that for a day every time he does something wrong. Or has to earn them back for doing thing around the house keeping his room and bathroom clean 

Third, do the same think with DH,  either he does parenting with SS or he loses being with you.