有人发钱咱就要,丢了工作咱再找.Nothing 2 worry about.

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