Hi Dosu,
There are currently 861 Kanji in the 人名用漢字 list.
Jinmeiyō kanji - Wikipedia, the free encyclopedia Should answer some of your questions.
I'm not sure what you mean by "I know there are multiple ways to write a name with several different kanji" though... If a person's name uses a certain Kanji then that is the only correct way to write their name. You can't just substitute in a phonetic equivalent Kanji if that is what you are suggesting.