A. Will this affect my admission in college during document verification or job at PSU ?
Yes, it would affect.
B. Does correction in his name is required or it can be ignored since all my details are correct?
Yes, it needs to be corrected and cannot be ignored.
C. You have your father's name wrong in the 10th and 12th certificate but your Transfer certificate has his correct name, this means that there has been an error committed by the schools or the boards while issuing your certificates
You can meet the Prinicipal or the head of the institution and get the offices to correct you 10th and 12th certificates.If they accept their mistake and accept your application,then good.
But the problem is many schools would not admit this mistake on their part and you may have to spend a long time following up the issues. Similarly, you cannot run around to the boards pointing out the mistake and spend too much time trying for the same.
In case of CBSE 10th or 12th an affidavit regarding the correct name will be asked along with your application.
So, first you need to have an identity card that has your father's correct name, this can be his Aadahaar card, PAN card, Voter's ID or Passport.
If you do not have one with your correct father's names, then you need to get a name correction affidavit done by a Notary, wherein you will have one legal document that would be accepted as proof for your father's correct name.
You need to publish this name correction or proper name of father in two newspapers one English and one in vernacular language.
Then draft a covering letter addressed to the respective boards (10th and 12th) requesting the service you require and importantly forward this via the prinicipal of the 10th and 12th school/college.
The school would forward the application to the respective boards and then in around 30 days it should be ready, you can collect it. It usually takes around 30 days.