"Is DNA evidence submissible in court to convict a man of a serious crime?" Absolutly! You hear about just that on the news every week.
"Is DNA evidence submissible in court to establish a man's paternity?" As far as I know, this is the only way to establish paternity if the person is wondering if the child is really theirs or not.
"The fourth part of my question has to do with birth certificates. Is a birth certificate considered an official government document and does it carry the same importance under the law as any other official government document particular in reference to perjury?" She can not list him without his concent. However if she needs any kind of government assistance she will have to list the baby's father and they will insist on a DNA test. If he is found to be the father then he will be on the financial hook for 18 years.