It good to here that your DP is getting help.
You have a very caring DS there. My oldest is also 4, he too is very sweet & caring. When the Steve Irwin died, Evan seemed to understand & get a bit upset but then would ask why he didn't go to hospital to get fixed. So I think little bit of him knew but then the rest of him didn't understand why he couldn't be fixed.
Every Child would be different but I think it would be safe to say that your DS would understand that the baby is gone & that makes you & your DP sad. He may not understand "why" Jesse is gone. But really he doesn't need to understand that part, he can see your sad & that is enough for him to know you need those extra long hugs.
Little boys are so sweet. Evan cries if I cry, he cried when Glenn was a baby & had to have his needles, he cried when Isla was crying & I told him it was because she got a fright. He hates to see his loved ones cry.