Tackling the feelings of incompetence and anxiety - You have enough runway. You'll manage for couple of months. It'd be alright. Just control your expenses. Also, remember that you are not your job. It ended, not your life.
Telling you were laid off should be fine. Be honest and don't lie that it was about insufficient pay.
Job market current status... Well, for which kind of role?
For the rejection - Send a thank you note (at least to HR) if they weren't rude and that you'd love to be considered for any relevant role. Then reflect on what may have gone wrong. Keep applying while you work on it.
Hope these help