You have asked a question that affords no straightforward answer.
In a way, the answer would be no, it isn't necessary for a person to have complete faith before entering Islam, which is by formally reciting the Shahadaah. However, a more important thing to bear in mind is that the new Muslim does not have any mala fide intent while doing so, and that he ...
Yes, it is true, If a person has Iman (belief) and he didn't shirk, he eventually enters paradise.
As said here
None shall enter the Fire (of Hell) who has in his heart the weight of a mustard seed of Iman and none shall enter Paradise who has in his heart the weight of a mustard seed of pride.source
'Ubadah bin As-Samit (May Allah be ...
You may also like to look at here which says:
Muslim has a general meaning (submission to Allah) and a specific meaning (commonly used now by people around the globe).
The specific meaning still has a large scope as it starts from apparent submission to Allah with minimal state of confession in uniquness of Allah and that the holiness Muhammad --peace be ...