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Why do they use sometimes small version of katakana ヵ 'ka' in a word 二ヵ国語 (nikakokugo -bilingual)?

For me, more intuitive form could be 二か国語. Katakana characters in small forms other than ya, yu, yo, shi + vowels should not even exist. ;)

Is there any significant reason for it?

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Main question: Why do they use sometimes small version of katakana ヵ 'ka' in a word?

The small ヵ here actually derives from the kanji 個 or 箇, used as the generic counter for things. This was abbreviated to 个, and in turn, this became the regular-sized ケ or small ヶ seen in words like 一ヶ月. Since the counter in these contexts is read as ka, this abbreviated kanji form ヶ came to be written using the alternate small katakana ヵ, probably to match the phonetics, and possibly to avoid potential confusion in readings with the actual katakana ケ.

So the reason would be gradual historical development. (As with most odd things in languages.)

Side question: Why do katakana characters in small forms other than ya, yu, yo, shi + vowels even exist?

There's the historical development reason above, for small kana used in Japanese.

Past there, kana are actually used for more than just Japanese -- they are also used to write Ainu. Ainu has final consonants, and the small-kana variants are used to indicate these. For instance, the word sir in Ainu means "place, land, country; mountain", and this is spelled in katakana as シㇼ with the small ㇼ to indicate the final /r/ sound. See this Wikipedia article for more about small katakana used to write Ainu.

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