I haven't read any of the other responses. But if this were on my plate I would not allow it.
THis is what I would do. First explain to him briefly, "___________, over then next few weeks we are going to practice being patient and taking turns, and if you whine or cry, you will go to your bed until you are happy."
So... what you do is when something hits his button. You immediately stop what you are doing and teach him an appropriate response. For example, sister is cuddling with you and he want to too. You say, right now it is sister's turn, if you want a turn you say, "mom, when sister is done may I have a turn cuddling?" If he cries, pick him up and put him in his bed (hopefully he is still in a crib, if not put him in time out somewhere, if he gets out put him right back. if that goes on where he is not staying - confine him to a high chair - a crib is a better choice in my opinion.)
Then you tell him, "WHen you are happy, i will listen for you and I will come and get you out." Then do what you said. Once he is quiet and seems calm (could be a long time but that is okay - it will be a good lesson he will not want to revisit) go get him and ask him why he was in there. Have him ask you the kind way to cuddle with you. Then cuddle.
The second he whines, teach him an appropriate response even if it is disapointment that is okay. We all get disapointed but we need to know how to act and how to talk to others through it.
After about a week of TRAINING him to say the positive thing, and he is still whining. I wouldn't say anything to him the moment he whines just pick him up and put him in his bed. And tell him you'll be back when you hear he is "happy." Or you can have him tell you when he is happy. Stick to it, be very consistant at it for a few days and I am telling you he will be living in his bed and won't have any other option but to improve his communication skills.
Also remember he has only been on the planet for 3 years. You need to train him and teach him every detail.
If you have any questions I would be happy to help. I hope this does help.