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.
這個軍隊具有一往無前的精神，它要壓倒一切敵人，而決不被敵人所屈服。不論在任何艱難困苦的場合，隻要還有一個人，這個人就要繼續戰鬥下去。
《論聯合政府》（一九四五年四月二十四日），《毛澤東選集》第三卷第一零三九頁
