My son went through a door slamming - actually, since he is 15 now, he has been through several door slamming phases. My solution - threaten to take the door of its hinges. No bedroom door, no slamming. Anyhoo, since I had once taken all the interior door knobs off, after he locked my out of the playroom when he was 4, he understands that I was serious and the door slamming stopped.
At 5, your daughter is developing her reactions to frustration - you need to correct the unacceptable behavior and teach her a reasonable verbal response and appropriate outlet. It takes time and patience, but she will get it.
Let the two old take a break from potty training - he might not be ready yet. It is okay, he will get it also when he is ready. Mine got the pee thing fairly easy, but would literally get his own pull up on to poop. Drove me crazy - until I just let it go - he knew how to poop on the potty - and he did when he was ready.
Give the baby a kiss from me - poor little one to be sick.
When I feel like a rotten parent, I stop and look around and count my blessings. I have been able to provide a nice home, food, clothing, stability, and a moral compass to my son. I remind myself that while I love him unabashedly, that my job is to raise him into a responsible adult. If he is not happy with my decisions, it is okay, because I have 30 years more life experience than him and I really do know better.
Take a deep breathe.
You are good parent.