Change your self talk! Re-read your entire post and notice how HARD you're being on yourself. You're stressed, it's okay! But getting upset about being stressed is not going to calm you down.
You can just tell your boss you are going through some troubles at home - they don't need to know any details about it. Say you are working through it, things are looking up, and you do not feel over your head anymore, you were just going through some personal issues at home. Then leave it at that. Your boss does not need to know details because he is not your therapist.

Just because he talked to HR doesn't mean it has ruined your job prospects at all.
Maybe moving to another section of the company will be a good thing, because then you'll get away from that mean coworker.
Show up to work a little earlier in the coming few days or week. Then, once you get to work, relax.

My friend was a chef at a restaurant and he hated his job, he said he was so stressed out because it was very fast paced and he had people yelling at him from every direction while he was cooking - he said he finally started moving slower. He started being more present, and literally moving slower. Instead of dashing around the kitchen in a fit of anxiety at a fast speed, he moved slower with every single movement he made.
He took a couple seconds between each movement and deliberately slowed down, which made his mind slow down as well, it allowed him to actually concentrate, and the people who were shouting at him, stopped.

So this is what you should do while working. Slow down. Block out all the external nonsense, and focus on you. Just you.
Whenever you get stumped and can't figure something out - take a few seconds, breathe, think about it, consider your options, and give yourself a little confidence boost while you're at it. Even saying something simple like, "I'm very intelligent." or "Nothing is permanent. Even this situation." can calm you down and help you to feel unstuck.