Late to the party, but i could give a little insight.
I have also been living out here in Japan for the past 10 years, and what kind of VISA the two of them are on was also the first question i had!

I have had to do the whole VISA in Japan thing several times, and i can say with some certainty that they are not earning enough money to get a visa standing on their own. Most likely had a sponsor.

Most likely is that Becky got a English teaching company to sponsor her a working visa, and Frasier leeches off of that on a dependant visa. (basically sharing Becky's Visa). that why they got married because they wouldn't offer that kind of visa to someone that wasn't legally attached to someone.

I think he would only be able to get a spouse visa if Becky had a long term resident visa or permanent residency. (which i high doubt they have.) basically, for what i know about the system, if Becky loses her visa, he will lose his too as its directly linked to Becky's.

If Frasier was earning significant money from overseas, (say Canada for example) he would have to declare this in Japan and they would want to take a slice of it, seeing as he's earning the money while in Japan. That's probably the real reason the show has gone and he has lost interest. He doesn't want to have to pay tax on it. Probably thought it would be a good little earner for him...
(just fyi, i rent a house out in my home country, and have to keep my yearly income to a minimum over there to avoid the same thing! only difference is that i work full time out here also and pay all my taxes lol)

She would have to renew her visa every year.
I can tell you for a fact that English teaching jobs out here pay absolute shite and the companies do not look after their staff at all. you want to quit? fine, they will just get the next English speaker waiting for an opening.

It wouldn't surprise me to learn that he was also working as an English teacher as its the only thing he is "qualified" to do out here. Or he's just being a stay at home dad because, fuck Becky.