Second, the registration card is for you to take to the personnel bureau for file keeping. There is a declaration period on it, so it's better to do it within the time limit.
Third, the four-party agreement you mentioned should be a three-party agreement. One for the company, one for the school and one for the individual. This is to count the employment rate of the school. Now that you have graduated, sign a labor contract with the unit. All three parties are useless.
Fourth, the problem of cadre identity and professional title is the same as you think. It's useless not to go to public institutions or state-owned enterprises and not to take an examination of civil servants.
There is a special topic about tripartite agreement on the capital talent network, which is very clear. You can go and have a look.