I didn't read all your responses, just wanted to add my quick "2 cents". Make sure, especially for your 2 year old, but still very important for the 7yo too, that they get outside at least 20 minutes a day (not counting whatever they do at school). I know you said you are sooo tired of the snow, do not hibernate indoors - you will all end up with cabin fever and the kids will act out. Just bundle up and go for a quick walk, I have always found this necessary for my daughter, who is now 7.
For your 2 year old, a time out should be 2 minutes (1 min/year), for both of them, explain that you expect good behavior at school, define what that means to you, and tell them what the consequences will be beforehand, and of course, FOLLOW THROUGH with them! Your 7 yo is young for 3rd grade, many of the kids will be 9yo, the rest will be 8; there is a BIG difference in maturity and the teacher will be expecting him to act like a 3rd grader not a 7 year old. I don't think you should punish him at all for talking in class, just explain that every teacher he will have will have a different personality, some of them will be really great, some will be not so nice, some will be good teachers and others won't be as good. Tell him the teacher doesn't hate him, but the teacher does expect him to sit and listen and not talk to others because they need to listen too. Also, my daughter has a really hard time being corrected, if the teacher told her to stop talking, she may feel she was being yelled at or in big trouble without the teacher even raising her voice. Because your son is so intelligent, he may come across as very mature in many ways, but he will still be, emotionally, a 7 year old. Second graders jabber away in class, whether the teacher likes it or not, in third grade this is not tolerated.
I think you are doing fine with your kids, it sounds like your 2 yo is acting like a 2 yo. Get a breath of (way too chilly) fresh air, and just keep teaching them what you feel is right and wrong and keep showing them you love them.