Most of the katakana names have pretty unambiguous translations, although the "ギヤマン" in "ギヤマンヘアクリップ" sort of baffles me. I know Diamond in Japanese is ダイヤモンド, but where'd the ギ come from?
I wouldn't leave "Seiryu" untranslated, unless other FF games do. Something like Bluedrake Scimitar?
蛇矛 is an actual thing
, it's actually pronounced either "Dabou" or "Jabou". Food for thought I guess.
方天画戟 is a Romance of the Three Kingdoms reference, romanized as "Houtengageki" (Japanese, I think the Chinese romanization might be "Fangtianhuaji" but I'm not sure).
"Cross Shuriken" for 十字手裏剣 is good.
風車 is actually more like "Windmill".
I think 円月輪 is another word for chakram
風斬り could also be translated as something along the lines of "Wind Blade".
"Manji Shuriken" and "Fuma Shuriken" work.
折り鶴 literally means "origami crane". If the weapon in question isn't a small paper bird, then you can leave it alone.
不倶戴天 is kind of tricky: I'd go with "Fugutaiten", and if I absolutely had to pick an English name, "Mortal" or "Irreconcilable"?
法螺貝 could also be "Triton's Trumpet" (its English common name), which I personally think sounds cooler.
腕輪 in all of its appearances translates closer to "bracelet" for what it's worth.
安式防具 and 甲型防具改 are also kind of tricky...
烈火の腕輪 - I'd go with Inferno Bracelet.
天雷の腕輪 - Thunder Bracelet.
防弾チョッキ means bulletproof vest.
銀縁メガネ are specifically silver-rimmed
You could probably get away with calling a ハチマキ a hachimaki.
I like your translations for 炎の指輪, 冷気の指輪, 雷の指輪, 怒りの指輪, and 水の指輪. They feel a lot less generic.
盗族の小手 is more like Thief's Gauntlet.
I'm pretty sure チョコボックル is more often translated as "Chocobuckle".
なんとか???, I believe, is more like "Something???"