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龍 or Dragon

Each line contains one word, that as a Chinese radical, contains one more stroke than the word from the prior line.

龍 or Dragon

Each line contains one word, that as a Chinese radical, contains one more stroke than the word from the prior line.

_{See this Wikipedia article for the complete list of radicals that powered this answer.}

DRAGON

Reasoning:

Dragons are mythical creatures in far eastern cultures. This relates to the title and latest hint. Additionally there are little hints to this with corpse, fire, fighting, I take villages and teeth, but these are likely a coincidence.

However, I'm having trouble relating it to the poem as a whole. So far, the only relationships I've identified across lines are:

The ending rhyme (given).
The opening rhyme for each line (e.g. te, fi, bam, gol, etc.).
The letter e is present in the last word of each line, excluding the second line, and including the hint about the non-existent line 17.

I can also say, without a doubt, that the next line will:

Start with the same letter and opening sound as dragon (e.g. dr, dra or drag).

It's implied that:

The word dragon is related to the word flute in this puzzle due to the 3^rd hint. This would also mean that the first word of each line determines the last word of the line that follows.

Dead Ends

I've evaluated the following, and unless I'm overlooking something (which I likely am), I haven't found anything that relates to my chosen word:

• The number of spaces.
• The number of letters in each line.
• The counts for each letter, in each line.
  • Including individual counts for vowels and consonants.
• The counts for each letter, across each line.
  • Including individual counts for vowels and consonants.

I’ve also reviewed examples for about 10 different styles of poetry that are popular in Asia and the Pacific, and this doesn’t match any of them, so that was a bust too.

龍 or Dragon

Each line contains one word, that as a Chinese radical, contains one more stroke than the word from the prior line.

Line by Line

"You insist I should take one"

The word one is represented by the radical 一, which has 1 stroke.

"Ten I shall take as I won."

The word ten is represented by the radical 十, which has 2 strokes.

"Tendencies are like a corpse"

The word corpse is represented by the radical 尸, which has 3 strokes.

"Fire spreads over entire course."

The word fire is represented by the radical 火, which has 4 strokes.

"Fighting as they found a hole"

The word cave is represented by the radical 穴, which has 5 strokes.

"Bamboos grow on land as whole."

The word bamboo is represented by the radical 竹, which has 6 strokes.

"Bamboozling, I take villages"

The word village is represented by the radical 里, which has 7 strokes.

"Golden signs lie over ages."

The word gold/metal is represented by the radical 金, which has 8 strokes.

"Golfers shall smell a fragrance"

The word fragrant is represented by the radical 香, which has 9 strokes.

"Horses show what they will glance."

The word horse is represented by the radical 馬, which has 10 strokes.

"Horizons pointed by a deer"

The word deer is represented by the radical 鹿, which has 11 strokes.

"Black as Guinness, oh my dear."

The word black is represented by the radical 黑, which has 12 strokes.

"Blocked by traps for tiny mice"

The word mouse/rat is represented by the radical 鼠, which has 13 strokes.

"Arranged evenly like dice."

The word even/uniformly is represented by the radical 齊, which has 14 strokes.

"Arrogant as they show teeth"

The word teeth is represented by the radical 齒, which has 15 strokes. This is actually the only radical with 15 strokes.

"______s those I hid beneath..."

The word dragon is represented by the radical 龍, which has 16 strokes.

Final Note

This answer was heavily driven by the answer given by @ExtraFishness, so make sure you upvote theirs too.

The only disconnect with their answer is that they focused on Japanese radicals. The Japanese radical for dragon is 竜, which only has 10 strokes.

Partial Answer

The word we're looking for is:

DRAGON

Reasoning:

Dragons are mythical creatures in far eastern cultures. This relates to the title and latest hint.

However, I'm having trouble relating it to the poem.

So far I've evaluated the following, and unless I'm overlooking something (which I likely am), I haven't found anything that relates to my chosen word:

• The number of spaces.
• The number of letters in each line.
• The counts for each letter, in each line.
  • Including individual counts for vowels and consonants.
• The counts for each letter, across each line.
  • Including individual counts for vowels and consonants.

I’ve also reviewed examples for about 10 different styles of poetry that are popular in Asia and the Pacific, and this doesn’t match any of them, so that was a bust too.

So far, the only relationships I've identified across lines are:

The ending rhyme (given).
The opening rhyme for each line (e.g. te, fi, bam, gol, etc.).
The letter e is present in the last word of each line, excluding the second line, and including the hint about the non-existent line 17.

I can also say, without a doubt, that the next line will:

Start with the same letter and opening sound as the word we're looking for.

The word we're looking for is:

DRAGON

Reasoning:

Dragons are mythical creatures in far eastern cultures. This relates to the title and latest hint. Additionally there are little hints to this with corpse, fire, fighting, I take villages and teeth, but these are likely a coincidence.

However, I'm having trouble relating it to the poem as a whole. So far, the only relationships I've identified across lines are:

The ending rhyme (given).
The opening rhyme for each line (e.g. te, fi, bam, gol, etc.).
The letter e is present in the last word of each line, excluding the second line, and including the hint about the non-existent line 17.

I can also say, without a doubt, that the next line will:

Start with the same letter and opening sound as dragon (e.g. dr, dra or drag).

It's implied that:

The word dragon is related to the word flute in this puzzle due to the 3^rd hint. This would also mean that the first word of each line determines the last word of the line that follows.

Dead Ends

I've evaluated the following, and unless I'm overlooking something (which I likely am), I haven't found anything that relates to my chosen word:

• The number of spaces.
• The number of letters in each line.
• The counts for each letter, in each line.
  • Including individual counts for vowels and consonants.
• The counts for each letter, across each line.
  • Including individual counts for vowels and consonants.

I’ve also reviewed examples for about 10 different styles of poetry that are popular in Asia and the Pacific, and this doesn’t match any of them, so that was a bust too.