There in fact is no such thing as a Shinma. The word itself was made up by Kakinouchi, so no there aren't Shinma in legends (although some are based off of folklore). The word Shinma consists of two Kanji: Kami - meaning a god, and Ma - meaning a devil. Japanese Kanji can have a variety of pronunciations depending upon the word, or how various kanji are grouped with other kanji. The kanji for kami is pronounced "shin".
The word itself gives definition to what a Shinma is - a being which is both god, and demon. Shinma are grouped together in clans. In the OAV they point out that Larva's clan is a renegade clan, in the manga there is clarified to be a difference between Western and Japanese Shinma (Larva being from the western sect). The Japanese Shinma lay in slumber, probably for the rest of eternity. It is Miyu's task to return the Shinma who do not sleep to the darkness. This also means that eventually Larva will have to be returned to the darkness as well.

Um... yes and no. The word for vampire in Japanese is "kyuuketsuki" consisting of three kanji, which directly translates into sucking-blood-spirit. (If I'm not mistaken this word came directly from China). However if you look at the title, the third kanji has been changed by Kakinouchi to be princess, so the new word is sucking-blood-princess (pronounced "Kyuuketsuhime"?). This is a nice little play on words that the Japanese occasionally like to do, unfortunately it gives translators nightmares. AnimEigo decided to keep the translation for princess when the released the OAV. The manga however was still translated just to Vampire Miyu.

Vampire Yui is sort of a spin off series of Vampire Miyu (the third kanji is also changed for Vampire "Princess" Yui, but I'm not getting into that again). Yui's father is a Shinma, and her mother has heritage from Shinma, and blood from Miyu, so it's all sort of tied in there. The story of Yui is totally set apart from that of Miyu, although she does make an appearance in Shin Vampire Miyu.

That's sort of debatable. There are a lot of things which are hard to interpret from Japanese. There is a lot of confusion over the names and such. No on is the almighty when it comes down to porting translations into English for which there is no equivalent. You may notice that my site is now called Kyuuketsuki Miyu, instead of Vampire Princess Miyu (but I guess that pronunciation may be wrong too). There are a few points here and there about various details, but nothing major. The point still comes across, regardless of certain trivial translations. Simply put, my Japanese is VERY poor, I'm not getting into it.