You are so right. He is dealing with his new sister. And he's only 3. therefore I would go with the "give him what he needs" philosophy. He's not old enough to reason through his behavior, he is just acting out because he feels he needs more attention than he did before.
I would be inclined to have dad sleep with his son until he falls asleep (the son), then he can return to your bed. It would help to keep the little guy in his own bed, and you would both be more apt to sleep. I don't think it will last all that long. I think if he sleeps more at night, he will be more apt to get his naps in during the day. When we get overtired, we tend to get on overdrive and we can't sleep. It's the same for kids. The only other thing I can think of about nap time is that I'm wondering if he is getting less exercise now that there's a baby in the house. Maybe if you can put Gemma in a baby carrier or something, and go for a walk with Jacob before naptime, it might also help him to be ready to relax and nap. ??
You are probably doing this, but when you diaper the baby, have him stand with you and help. He is 3, and that makes him big enough to help. He can pass you the baby wipes, and the diapers, and he can help with the tabs. If you use real diapers, he can probably hold the pins for you, and pass you the things you need. Then he can get self-esteem big boy kudos for being a part of the "big people team".
If you are nursing, you can read stories to him while you are nursing. The little one can eat, and you and Jacob can have some group time at the same time. If you are bottle feeding, your hands are a little more tied up, but moms get pretty engenious about accomplishing things with only 2 hands. Or, maybe JAcob can hold Gemma and feed her for a while. He'll get bored quickly, though, so you have to be prepared when he pushes the baby off his lap. (That's normal, not abusive behavior -- but you have to be prepared and ready to catch her -- or have him sit on the floor leaning on the couch or something to feed her so he's not up high if he pushes her away. Hopefully you can tell him to let you know verbally when it is your turn, so he doesn't just push her out of his way when he's done.)
And crying and yelling ? Well, he's acting like Gemma, because whe she cries, she gets attention. When she cries you can remind him that she doesn't yet know any words, and that it's going to take a few years, but it's up to you, Dad and him to teach her words. So when he wants something, he has words and he needs to use them. You can always tell him that you can't hear him when he's yelling, and he needs to talk quietly so you can hear him. (I know, it doesn't make sense, but it works if you ignore him until he tells you what he wants.)
Some of this behavior will simply settle down over time. What's really important is that he know he is loved, and you keep your sanity as you go through this time period in which you are likely exhausted as well !! The way I look at it, is that whatever works for Mom is probably good for everyone. That way Mom is more relaxed and that makes everyone in the home more relaxed. :-)
Congrats on the new baby ! And know that while it's tough now, it WILL get better !!