"Begins to cry" and "starts to cry" have the same meaning and are completely interchangeable. "Starts crying" uses a different verb tense than "begins/starts to cry" and implies that the crying will continue in the future. There is a subtle difference between "starts crying" and "begins/starts to cry." Native English speakers never say "he has not a problem" because it sounds arcane. "He hasn't got a problem" has the same meaning as "he has not a problem", but it's bad English. Instead, we would say "he doesn't have a problem." "Have got" and "do have" have the same meaning, but "do have" is more common among native English speakers. Many English as a foreign language books teach the "have got" structure, which we professional English as a Foreign Language teachers loathe.

Style point: it's a little rude and selfish-sounding to use "wanna" when you're making a request to people you don't know. Use "I would like..." My advice is simply to never use "wanna". Ever.

For the Begins to cry VS starts to cry VS starts crying, they are pretty much the same thing. It depends on the context though. They are all present tense though, so if you use them, most likely it will be bagan to cry or started to cry. As for the "He has not a problem (or he hasn't got a problem) VS He doesn't have a problem." thing, it would be 'He doesn't have a problem." If you say that he hasn't got a problem or that he has not a problem, people will be able to grasp what you are saying, but gramatically it doesnt make much sence.