I was Team Cap before I saw the movie. I am still Team Cap. As for your argument:
1) I really feel for Tony at the beginning of the movie because things just keep getting worse for him, so I understand why he'd be mad.
As I said earlier I really felt bad for Tony throughout the film. Until the very end he was doing everything he could to protect Steve and the others. He was doing everything he could to reason with Steve.
2) I don't think Cap wants to break up the Avengers. He just don't want to sacrifice his ideology no matter what the outcome is. Steve has never been any less than real to himself and to others. Signing the accord would go against everything he stands for.
I don't think either of them wanted to break the Avengers up, the problem was Steve never offered a valid alternative. He refused to trust anyone else but expected everyone else to trust him. It was like Rhodey said, Steve's stance was dangerously arrogant.
Natasha and Tony understood that it was going to happen regardless of what the Avengers decided, but if they signed they at least would have been able to have input into what the accords entailed.
3) Keeping the secret may not be a wise thing to do in hindsight, but I believer very strongly that it's because Steve viewed Tony as his friend, and knowing how much it would hurt Tony he decided to keep it a secret. Besides it's not like Tony asked Steve and Steve lies about it. Keeping mum is not the same as telling a lie.
This stance is a bit of a reach considering that even Steve says he was sparing himself more than he was thinking about how much it would hurt Tony.
And lies of omission are a real thing. If my friends wife is cheating on him and I know about it, not telling my friend is just as bad as lying to him about it. Keeping things like that from people rarely keeps them from being hurt because they usually find out the bad secret anyway AND they find out that someone close to them knew but never told them (which is also hurtful).
It was also a bit hypocritical of Steve considering how upset he was that Sharon Cater was an agent tasked with keeping tabs on him.
Too often people conform because it's the path of the least resistance and make things ok on the surface. It's refreshing to have a hero who is willing to give up everything and do things the hard way because of what he believes in. Tony chose the former and therefore I have more respect for Steve in CW.
I'd agree with this except Steve was just being unreasonable, he wasn't willing to comprise to work out something that would be mutually beneficial. As I said earlier in this post, Steve refused to trust others while at the same time expecting everybody to trust him.
There was never a point where he offered an alternative, like making sure the Avengers have an U.N Liaison, or making sure the Avengers were completely transparent with the teams activities. His stance was that things were fine as they were even though they had just accidentally killed a bunch of people in this movie and they (as a team) were responsible for all sorts of damage and deaths during Age of Ultron.