However.... everyone has different parameters here - and that's where the confusion can start. I think that tiny children who are shy should not be expected to say thank you etc - but that it is quite right for the parent to say it to the person for them. As they get a little older, you could start to whisper to them (perhaps bending down to them "shall I thank Granma for the present or will you?" Or - "Let's thank Granma together" and playfully trying to do it.) Sensitive adults should see that your children are shy - but I do think it's important that they are seen as participating in good manners in some way - it could be that you pick each one up as you thank the person - they are thus participating via you.
I think the whole kissing and hugging issue is very sensitive and very difficult. Part of me (despite a part Jewish part Italian very huggy and kissy family) thinks it's awful to submit children to that ordeal. On the other hand, on the rare occasions when I brought my sons (brought up in the UK where people stand about a mile apart when expressing gladness to see each other) to NYC to see the family, I figured it wouldn't kill them to be kissed and hugged and thrown around and generally passed from hand to hand by my enthusiastic relatives. OK - they were a bit shocked by the end of it - but they survived - and they learned something about the other side of their family.
So there's a lot of cultural issues here - very hard to navigate.
As for the play thing.... stop talking and start doing!! You need to keep them fairly close during those times that such squabbles are most likely to develop and get right in there and show them how to play. Remove the toy from the offender. Give it back to the one who had it stolen. Take the offender away from the situation and go peel carrots with her in the kitchen. Accompany her back and hand her a new toy - if she insists on the old toy, take her back to the kitchen. If she cries and screams - let her. It is ok for children to cry. There's a bit of an AP (attachment parenting) thing against crying which sneaks in, inappropriately, to the post-baby years. Letting a baby cry it out in a crib - I also could never do that and don't think it's healthy. But as the child grows, if she needs to cry out of frustration or anger, that's ok. She has no other way to express herself - and does not need another way until she grows into an ability to start to express her feelings - and that kind of self awareness should not come at such an early age. Many of us fear our children's strong emotions, feeling somehow we are failing them if they rage or cry. Thus so many parents try their hardest to teach their children various skills so that they can speak and name their feelings. But I would say that this is actually far more harmful than letting the child cry or rage. By bringing her precociously into self awareness (which is what one needs in able to name ones feelings) then one is shortening the pre intellectual stage of consciousness of the young child, the stage of oneness where there is no true sense of "I". These are the years of working and learning via the physical body - and physical responses from the child such as crying and screaming are part of that.
As for the whole apologizing for hitting her sister thing, I would just say simply but firmly to the child in such situations "No - that's not ok." Remove her. Then later on, help her do something nice for her sister (redemption - always redemption). At 3 1/2 she is too young to do this on her own. So you say "We're going to draw a nice picture for your sister. She felt bad before when you hit her." No recriminations, no guilt, no therapy-speak, no fuss. Simple fact - this happened, this how we make it better. Even if she does not cooperate - you do it. You are still powerfully connected to your little ones and what you do influences them. You give the other child the picture in the "offender's " presence - and in her hearing say "This is a card from X. " You don't even need to explain. They know.
I think part of your quandary, Member X, might be a missing piece from the modeling/ discussion thing - it's the physical aspect. Little ones need to be picked up, handled, moved from one place to another, removed from situations, held when they scream etc etc. This is scary - and hard. But without it, one is forced into discussion with them - and we all know how useless that is. And you have discovered the limitations of modeling - it is important what we do, but when the child is in a state of anger, is out of herself, she can no longer copy our behavior. She needs the parent to provide a physical response.