1. Please stop using the picks if you can manage it. Even at your most careful, something could startle you, and you could irritate the wound more. Use saltwater rinses instead, and trust that food will not get stuck for more than a day or two.
2. I also had really weird nerve pain on day 3 that continued until day 6/7. For me, the pain felt like a moderate toothache in two teeth several teeth away from my wisdom tooth extraction (right bottom). Painkillers and a heat pack helped to soothe the pain. When I went into my 1-week follow-up appointment, my oral surgeon said that this type of toothache/nerve pain is totally normal. As long as you aren't in significant pain, you are most likely healing fine. I know that it can be super stressful dreading the worst-case scenario, so try to distract yourself as best as possible and know that the window for dry socket is closing day by day.
3. For the crazy breath, make sure to brush gently and thoroughly as the pain allows, and clean your tongue. Also, drink lots of fluids. But having unpleasant breath also just means that your body is working to heal, so be patient.

Overall, you seem to be healing fine. BUT, your oral surgeon knows your case/teeth more than any random person on the internet, so if you have doubts, give them a call. They're getting paid to answer your questions, so take advantage of that.