In reply to:

1 to 10 - unique(*see below) words for each
11 - ten-one
12 - ten-two
13 - ten-three
...
20 - two-tens
21 - two-tens-one
31 - three-tens-one

Apparently, the island of Taiwan is favoured as a likely point of origin for Polynesian peoples. I mention this because Maori uses exactly the same counting system as the one you listed - interesting, given Japan's proximity to Taiwan. Perhaps the similarity is not coincidental.