Sorry for the late answer and the text wall

1. Depends on the severity and how much they know about my avoidance habit. I would say sorry (as genuine as I can) and told them about my problem (or most likely make up a believable one that neurotypical person can accept) that made me unable to respond to them in timely manner. That it is wholly a Me problem and not a You problem. I would try my best to complete what they ask of me or help them with something else just to keep it even. This vulnerability talk will exhaust your mental energy tbh, some people will definitely have negative reaction and rightfully so, especially in professional setting. Tho some bridges have been burnt so bad I don't even try 🫠 Still, it's very much worth it, rather than spiralling down to depression town, sinking in the guilty mud trap.
2. I too wish I have an answer for this one. If you have the means, try finding out the reason. Maybe it's ADHD, maybe it's AvPD, maybe it's both, neither, or something else. So far, applying CBT (Cognitive Behavioral Therapy) has helped me improve, but it's not a bulletproof solution. My avoidance mostly comes from my tendency to be a perfectionist (which comes from past experience, upbringing, yadda yadda), so I try to accept that sometimes it is okay to be flawed, to disappoint, to not try my best for someone else's sake just to get their approval. Try to have healthy habits that improve mood/health. I also try not to spend too much time inside all day without actual human interaction, because I can definitely get more anxious when I haven't interact with real people (actual interaction face-to-face, not text/chat with people online) to keep me grounded.

Just keep on keeping on one step at a time.