For the most part, I think your son is totally normal with his screaming. Do take him to the DR for a good look over if you haven't already, just to be on the safe side. You also might want to consider taking him off the bottle totally. This could be putting air in his tummy and causing gas bubbles and pain.
As for the screaming it's self, this is, for the most part, a phase. He has found a voice this way and he uses it every chance he gets for everything. He also gets a reaction from it from other family members as well as you, especially when you scream back at him. Now it has become a great game of who can scream the most, the loudest, etc. And don't think that he can't pick up on daddy feeling stressed or nervous about his screaming as well which is great for your son because now he feels he has some sort of control over the both of you by making you scream and walk on pins and needles.
Start off by looking at him and saying, that is not how you talk to mommy/daddy, when you are ready to talk nice I will listen. That is not how you ask for milk/juice/food, when you are ready to ask nice, I will listen. And then give him a bit to ask properly. If he doesn't, go about your business till he does.
Consider trying to get him to talk instead of using sign (after he gets a bit older, have him do both at the same time), he could be trying to be verbal and this is how he has figured out a way to do it. You could also start teaching him time out (one min for now) for when he screams. Last but not least..You must ignore his screaming! Ya, I know, you said you did that already, but do it it again and don't give in this time. Don't talk to him, don't look at him, don't even be in the same room as him when he does it if that's what it takes for you to ignore it. And that goes for everybody. Just don't react-plain and simple!
It's a bad habit that he has devoloped and it is up to you to stop it. There is nothing you've done wrong that caused this either. He is just being a toddler!