It is amazing you can get the unemployeement benefit even you said you left the company in that way. I can't believe it.
1) the next employer will know that I was fired or not if they check with my former employer?
-- It totally depends on how your HR answer the background check questions. Usually HR will not say any thing else besides your start date and end date.
-- There is nothing to worry about this.
2) when I applied unemployment insurance, I was asked if I was laid off or fired, I said fired. will this become public record and any employer can see it when I apply for new job?
-- I am surprised you can get unemployeement insurence in this case. Are you sure?
-- Again, there is nothing to worry about.
-- Update resume, start looking. You will find a job.
这个军队具有一往无前的精神,它要压倒一切敌人,而决不被敌人所屈服。不论在任何艰难困苦的场合,只要还有一个人,这个人就要继续战斗下去。
《论联合政府》(一九四五年四月二十四日),《毛泽东选集》第三卷第一零三九页