S13 tries to walk out and I told him to sit down, we weren't through talking, so he tries to "perch/lean" on the couch...
"S13, please sit down properly and give me your attention.."
He finally kinda complies...but is still out of sorts.
I ask him to explain the importance of what the dog eats...he talks about the dog being a diet (that was a year ago).
I asked if there was any other reason...nope.
I asked what dog had been doing for past month...regarding pooping.
S13..."Oh..that..."
So, a month or two ago, it was the end of our weekend with the kids. DH planned a hike that he and I had done before -- easy meadow meander along a stream, no elevation, just getting outside in fresh air.
SD13 said she didn't want to go hike. So, we said That's fine, you are welcome to stay in the truck at the trailhead. She said no, she didn't want to do that, she wanted to stay home (alone!

) and do nothing, and then something about why do I have to go if I don't want to.
DH said that we were doing something all together as a family. She doubled down on not wanting to go, and not taking the compromise.
I let DH deal with it for about 20 minutes, and then he came and asked me for help, because she wasn't budging.
We agreed that some sort of "consequence" looked like it'd be necessary, but the big one we saw (taking back a birthday gift she just got) seemed manipulative.
We talked to her together and said we understood she didn't want to hike, and that was fine, and she didn't want to sit in the truck either, and that was fine, and she wanted to stay home, and that was fine, so if she chose to stay home, she'd be staying with me and doing some chores or HW. Staying home alone and doing nothing wasn't on the table.
She didn't choose any option and escalated. We left her alone to "think" but after ~10 minutes couldn't find her.
She was on the roof...

DH and I agreed not to chase her up there. DH went back inside and was pretty steamed. He told me he'd told her to come down and she was not obeying.
It was just me and SD13, so I tried a lot of "tell me more about how you're feeling", but what I got back was "you never listen to how I feel" (so, in retrospect, that was a clue that she wasn't tracking any more-- I'd say "I really want to hear more about how you feel" and she'd return "you're not really listening to me")
I let her know that DH had asked her to come down to talk, and I could hear him coming back. She said she could talk better on the roof. I said if she wanted to avoid a bigger consequence, she would need to come down to the ground to tell him that personally.
She GRUDGINGLY and BARELY made it down to do so (which was a win) -- that reminds me of your S13's "leaning" on the couch and hoping that "counted" as sitting (trying to one up you).
Anyway, after another 20 minutes or so that felt like an eternity, it was clear that she wasn't engaging rationally. She did have one moment of clarity of "I get that when I say I want to stay home and do nothing, that sounds selfish", so we said Thank you for your insight.
We had to go to "Well, if you choose to stay on the roof and not follow instructions, and not choose an option, then we are going to hold onto Gift until you are mature enough to listen".
THAT got a huge response -- crying, huffing, stomping, down from the roof, following us through the house, "you never listen to me, you don't care about how I feel"... all of it.
So we said she had ten more minutes to decide. I put the gift box out in the middle of the room "to think about".
She GRUDGINGLY got in the truck and we left. We tried not to do any "punishment" with attitudes or "now you made us late" or whatever.
Here's the kicker...
We get to the trailhead and everyone eats some snacks. I assume SD13 is staying in the truck, but
SHE COMES WITH US
And thinks it's a great hike, and is like, "This place is amazing, I love it, I'm having so much fun"
DH and I just had to be like "Don't say anything yet... don't say anything yet... wait for it".
Later she apologized to me for her attitude. She said she thought it was going to be a mountain hike and she didn't want to do that.
I told her I forgave her, and it sounded like what would help her in the future was more information. She agreed that she could have asked for clarification about what we were doing.
I checked with DH and he said that she apologized to him later, too.
...
But... there are some very "13" things going on for both S13 and SD13:
-barely meeting requirements (wanting to be on top/dominate?)
-underwhelming response to "the whole situation" (i.e. "Hey, I love it here, I'm having fun!" right after a 1 hour meltdown, and S13's "oh, that")
-trying to walk out/leave/escape facing conversations about their choices/behavior/attitude
I think it's good that you got S13 to sit on the couch. Kind of like getting SD13 to come down off the roof to say that she wanted to be on the roof, and then later get in the truck. It telegraphs that you are still the adult in charge.
The dog/food issue seems more like "generic 13" versus anything weirdly specific, but as GaGrl said, could be a "sprinkling" of other things going on that flavor his "standard 13" behavior.
Also, if he suggests that he is hard of hearing... great opportunity for you to leverage that in ways that "ensure he can hear you OK". Not sure exactly what that would look like, but man, that is such an open door to "believe exactly what he says".
"Oh no, you had a hard time hearing my instructions... I'm worried... I'll make sure to _______ (sit right next to you? write everything out? other?) so that we don't have this issue in the future."
...
Would FFw be on board with getting S13 evaluated -- maybe rule some stuff out, so you guys aren't beating your heads against a wall if you don't have to?
...
You have a full plate... I know the feeling of reeealllllyyyy not having room for attitude shenanigans. Hope your mom and you have some better days.