Welcome, Mama! I'm sorry you're having such a hard time right now. I'm with the PPs that an evaluation for postpartum depression seems warranted, based on what you've written. It's normal for a new mama to be exhausted, frustrated, even short-tempered. But if you've crossed over into feeling angry at your partner and child, and your support system perceives that there's a problem, that's a serious thing that should be taken seriously.
Everything you describe about your baby sounds very normal. Waking every 3 hours? Normal. A lot of babies are still waking every 1.5 hours at 3 months old (both my girls were). Distractable? Normal at 3 months, now that baby is perceiving more of what's going on around him. Fussy and gassy? Common in young babies, and not necessarily a sign of something being wrong, or of the baby having a dairy intolerance. Poops and pees sound normal. Eating to contentment and then stopping is normal. (Consider the alternative: the baby eating until he's overfull and uncomfortable.)
I'd encourage you to see a therapist, and I'd also consider doing some bottles at night. They can be filled with breastmilk that you express during the day, so that you don't need to get up and pump at night. Or if that won't work, just use formula at night. It doesn't have to be forever- just until you feel more together.