Maybe it's a sentence about some kind of list, or some kind of database. I believe "We do not have his name in the database" makes sense. At least «У нас в ба́зе да́нных нет е́го и́мени» definitely makes sense in Russian.
No, it sounds unnatural. In your context, «Мы не знаем его имени» sounds much better. (Alternatively, «Его имя до нас не дошло» works too.)
No, this is not something we'd say in Russian.
I'd say «На́шего па́пу зову́т Бори́с, а нас зову́т по-друго́му».
But "leaving a name" came from people leaving calling cards and later business cards. So now we have a shortened expression meaning that their name is unknown. Strange saying without the social context, I guess. Here, unless we are making a list and don't have this person's name on it, this sentence is extremely awkward.
"There are also a group of several irregular "different-declension nouns" (Russian: разносклоня́емые существи́тельные), consisting of a few neuter nouns ending in -мя (e.g. вре́мя "time") and one masculine noun путь "way". However, these nouns and their forms have sufficient similarity with feminine third declension nouns that some scholars such as Litnevskaya consider them to be non-feminine forms of this declension" https://en.m.wikipedia.org/wiki/Russian_declension