Used to be $320, now $395. This is one time filing fee charged to the respondent. Otherwise, who file, who pay.
Joint petition is a special procedure for non-contested divorce. There are special requirements such as duration of marriage less than 5 years, no children, community assets less than $25000, community debts less than $5000, etc. It would still need to run the six months judicial cooling period but it is quicker compared to most contested divorce.