There are many instances where this happens in Japanese. For whatever reason, sometimes the Kanji is not used with a word.
Here's a few examples:
ありがとう (normal) 有り難う (kanji form)
よろしくおねがいします (normal) 宜しくお願いします (kanji form)
こんにちは (normal) 今日は (kanji form)
Off the top of my head there are four reasons why this happens (possibly more).
Kanji is ...
First of all, I am also no expert, but I have been looking at classical Japanese orthography recently and noticed that many of the "spelling-change rules" seem to follow the same logic as some modern Japanese's collocations/"slang".
For example the simplifying of words by seemingly merging sounds: わからない → わかんない。If you take けふ and pronounce ふ as hu not fu, ...
Regarding the nametag, it does clearly say れいな -- it's probably just the joins between the elements that are making it 'look off' to you? Perhaps seen from afar it would be clearer, as those small joins would appear less prominent?
As for pronunciation, Forvo is a good place to check for native pronunciation recordings. This individual says "Reina Miura" (...
There's a very slight difference between an English pronunciation and a Japanese pronunciation.
While the English pronunciation would be Ray-nuh, the Japanese pronunciation would be more like Re'ena. You hold the "eh" sound for a bit longer. The closest example in English that I can think of off the top of my head is the first "e" in "pepe the frog".
As for ...