Quite frankly, my advice is not to babysit these kids. But if that is not an option . . . .
It sounds to me like the kid needs attention. So incorporate him into your day. Let him help with his sister, let him help you with chores or cooking if you’re doing some (or let him sit at the counter and watch you). Take him for walks, play games with him, read him books. Kids at that age are sometimes not good at amusing themselves, as others have already said. And over-reliance on the Glowing Babysitter (the TV) is IMO not a good idea – the kid gets wired and hyper and his behavior does not improve.
“Continue” to hit you? Hitting you or anyone else is not okay. If he does it once grab his hands if necessary, look him in the eye, tell him firmly that hitting is not okay and if he does it again he will be in time-out. Then follow through. Every time he does it. Tantrums, screaming, crying be damned. Same deal for getting into stuff or taking the baby’s stuff. Failure to listen to you – look him in the eye, tell him that when you tell him something he will have the count of three to do it, or he will be in time-out. Give him a directive (politely, like “please pick up your toys now”) and if he doesn’t, count off “one – two – three” loundly and firmly and if he doesn’t obey – time-out.
Ask him at some point during the day – not at bedtime – “what would make bedtime easier for you? Because you got really upset last time.” You can guide him with suggestions: I will leave the door open this much, okay? We could read one story to you after you brush your teeth, would that be better? I can sit with you for some minutes after you turn off the light; how many minutes would be good? (Then you take up an kitchen timer to mark off that time and when it bings you leave, after one last hug and kiss). Get his buy-in on the plan during the day, making it seem like it’s his idea, and he may be more willing to follow it at night because he will feel more in control of what is happening to him. Worst case scenario, he screams for 10 minutes. Bedtime is not negotiable.
Remove yourself from the couch and take the baby with you. You don’t have to stay where he can crawl on you.
Fake crying: “Are you tired? Do you need to go to your room for a little nap?” Not threatening but with real concern. The answer will of course be NO. Then ask him what’s wrong. He’ll have to come up with something. Address that issue and then tell him, now I have to deal with the baby for a while but they we’ll [play a game/ read a book/ whatever].
Grabbing you: If he’s explaining it as trying to hug you, say “Oh! Well, you can’t grab me because I might fall down and hurt myself, but you can certainly have a hug anytime you want one, just ask!” Then give him a hug. If he grabs you again, tell him that you’ve already explained that he can’t do that because you might get hurt, and if he does it again it’s a time-out. Third time: time-out.
In my world, a kid who intentionally shit himself out of spite would find himself in a none-to-warm shower to be cleaned off, followed by a session of cleaning his own clothes with whatever assisance was necesary from me – TP’ing off the undies, rinsing them in cold water in a bucket, putting them in the washer, adding the soap, and turning the washer on. None of this would be done in a loud or punishing way, but the process would not be fun and my disapproval would be pretty apparent. Then he would find that he had lost privileges not as a punishment but because “I don’t think you’re old enough to do that. You pooped your pants and only really little kids do that.” Again, nothing obviously punishing or done in anger, but demonstrating that if you act like a baby you will be treated like a baby. I doubt it would happen many times.
If he can’t amuse himself and you’re the sitter, then you have to amuse him. Take him and the baby for walks. Take them to a park and let him run around. Let him/make him help with chores. Let him help you with the baby. (“She’s kind of fussy again. What can we do to entertain her? Let’s try to make her laugh!”) Bake cookies. Play games. Read books. I’m not directing this at you Roses, but ISTM that a lot of times people think “babysitting” is that the sitter sits on his or her ass reading a book while the kids take care of themselves. It isn’t, and they won’t. The sitter has to entertain the kids, not just make sure they don’t set themselves on fire.
ADA: Affection, Discipline, Activity. And consistency in all three. That’s how you do it. Or else you don’t babysit for them anymore, which as I said initially would probably be my choice.
) “What is the goal here and how can we adjust to make it easier on everyone?”
You can always make suggestions, but of course it’s up to her to make the final decisions. Maybe you could just tell her you were talking to a friend, or someone on a message board, who had a child who was having similar behavior problems (huh, and I just realized my bratty kid was about 14 months when his father died and now at 3 has a new baby brother, so there are a LOT of parallels here) who found that severely limiting television really helped, and she’s suggesting you all give it a try for a few weeks and see what happens. Then have lots of other suggestions ready for what he can do instead - you had some great ones two posts ago.