Questions about Student X2

Question 1: So, I arrived in China about 35 days ago, on a X2-visa. I'm currently enrolled for one semester, so according to my X2, I don't have to apply for a Temporary Residence Permit since I'm to live here for less than 180 days, right?

2: BUT, I do want a Temporary Residence Permit since that would allow me to enter and exit the country without restrictions. I've got everything I need for it except from that I haven't done the Health Exam. It has been more than 30 days since I entered China, but would it still be possible for me to apply? I mean, since I'm not really obligated to apply for it.

Thank you!