the calligrapher, Mr Ishikawa, who was a believer...

Mr. Ishikawa is fine. (The Japanese give the clan name first, and then the given, the personal name. So, Oda Nobunaga was the had of the Oda clan, whose given name was Nobunaga.) I have noted that the Japanese often do not use the common, English title of respect, and might simply write: " . . . by the calligrapher, Ishikawa, . . ."

No, it would not necessarily be better. To refer to his profession without the definite article is not wrong, but it would look awkward. However, I will note that this is a stylistic convention which is changing.

The locution " . . . who was a believer . . ." implies either that Ishikawa has died, or that he has ceased to be a believer in the religion. I suspect that you will know which is the case.

It could be written ". . . the late calligrapher . . ." and that would clear up the ambiguity noted above. I would point out that although there is no hard and fast rule, generally using "late" in such a context implies that the individual referred to has died recently, which might not be known to the casual reader. Louis Feldman was long considered the greatest authority on the ancient historian, Flavius Josephus. Mr. Feldman died last year. Flavius Josephus died more than 1900 years ago. Using "late" to refer to Professor Feldman would make sense, as that might not be known to everyone. Using it to refer to Flavius Josephus would be ludicrous. Therefore, the question would be whether or not Ishikawa died recently, or long ago; and also, whether or not this is clear from the context of the entire passage.

Thanks, Setanta, for your very detailed reply.