• 2 This afternoon, my cousin on my father's side called me.

• This afternoon, my cousin on my father's side called me.

  Or you can say “my paternal cousin”

• 1 He said that his son would graduate next year.

• He said that his son wouldill graduate next year.

  This is a subtle correction, but since he told you (in the past) about something that is still in the future, "will" fits better than "would."

• He said that his son wouldill be graduateing next year.

• (His son is a postgraduate at Anhui Medical University) His son’s girlfriend had found a job in Fuyang People’s hospital.

• (His son is a postgraduate at Anhui Medical University.) His son’s girlfriend hads found a job inat Fuyang People’s hHospital.

• (His son is a postgraduate at Anhui Medical University). His son’s girlfriend hads already found a job in Fuyang People’s hospital.

• (His son is a postgraduate student at Anhui Medical University) His son’s girlfriend had found a job inat Fuyang People’s hospital.

  For someone who is studying for a degree after their Bachelors, we typically say Graduate Student. For specific professions, like doctors or lawyers, it is more common to say "Medical Student" or "Law Student."
  
  Similarly subtle correction about the verb tense here, but since you're describing just one event (the girlfriend found a job), you wouldn't say "had found" here. If you're describing multiple connected events in the past, then "had found" would make more sense, e.g. "She had found a job, but decided not to take it."

• 1 His son definitely wants to work in the same city.

• His son definitely wants to work in the same city as her.

  This is grammatically correct, but you could also combine this with the previous and/or the next sentence to make it read more smoothly.

• For this reason, His son definitely wants to work in the same city.

• So he wants me to inquire about some hospital recruitment information in order for his son to apply for it.

• So he wants me to inquire aboucollect some hospital recruitment information, in order for his son to apply for itto the hospital.

  you could say "inquire about a position" or "collect information".
  
  For job applications, the correct phrase is "apply to [a workplace]" or "apply for [a position]". Since you only mention the hospital here, it would make more sense to use "apply to," and it doesn't hurt to be more specific and say exactly what he's applying to ("the hospital").

• So he wants me to inquire about some hospital recruitment information in order for his son to apply for itjob opportunities at the hospital for his son.

• So he wants me to inquire about somthe hospital's recruitment information in order forprocess so his son tocan apply for it.

  You're not "inquiring" about the "information" (the "information" is the result of "inquiring"), instead your "inquiring" about the "recruitment process".

• I asked him to give me a resume about his son and I sent the resume to my wife.

• I asked him to give me a resume about his sonhis son's resume and I sent ithe resume to my wife.

  unless he physically handed you the resume, it would be more typical to say you asked him to "send" it to you.

• I asked him to give me a resume about his son and Isend me his son's resume and I will sentd the resume to my wife.

• I asked him to give me a resume about his son and I sent the resumefor his son’s resume so that I could forward it to my wife.

• My wife used to work in a hospital, so she has contact with some hospitals.

• My wife used to work in a hospital, so she has contact withs at some hospitals.

  She could either be "in contact with the hospitals," or she could "have contacts at the hospitals"

• My wife used to work in a hospital, so she has contact withs in some hospitals.

• Since My wife used to work in a hospital, so shehe still has contact with somes there as well as other hospitals.

• 2 I hope that there will be some good news for him.

• I hope that there will be some good news for him.my cousin.

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