Japanese Guys: Should I ask him if he has a girlfriend via e-mail?

Wow you look so attractive and I do believe most of Japanese males would think so too. But please don't be such American as to ask directly via mail. We Japanese won't ask things directly as you might have known as Japanese is your 2nd language. Japanese males still would like to hold initiative specially on relationship, it won't give good impression even he has no gf and was attracted to you too. I suggest you to ask things (anything will do - Japanese culture or your concerns and make it in serious way) via mail and after having several advices from him, say "〇〇の件では相談に乗っていただいてどうもありがとうございました。今度お礼に お昼でも御馳走させてください” in formal way to keep proper distance. I think going lunch together for those reasons is allowed in Japanese business mannter. Then ask about his private life "お休みのときはいつも何をされているんですか？" to seek your answer. I think this way is very common way for Japanese girls to hint their interests. Hope you can have a good luck getting over cultural differences.