[jasonbvr]

If you are getting the work visa from your programming job, there is no need to worry about what type of degree you have. The only ones who want to see the degree are immigration officials. When applying to jobs, just send them a picture of your visa and say one of the degree can be obtained if necessary. Most likely they won't need it.

As far as hiring natives, indeed there is a big preference for native speakers. That doesn't mean non-Asians but rather just from a country where English is an official language. But, I have heard of half Japanese, half Americans being turned down by Nova (the eikaiwa that just went under) because half their childhood was in Japan. Again however, you are not looking for regular work as a teacher just part time. You should be fine as long as you really emphasize fluency in English.

As far as math tutoring, that would be something Japanese students would go to jukyu for (cram school). It would require fluency in Japanese. There are some schools that teach math in English, but that would be a full time gig, most likely at a private school.