Your first question: the verb for death is shinu. When you take a verb and turn it into a noun, you change it from the U-form to the I-form, so shinu becomes shini, making shinigami.
Also, when you stick another word onto that I-form of a verb, often the other word changes in sound slightly so that it is easier to say. There are standard rules for how words change based on their sounds, and you'll pick them up quite easily if you continue reading. Anyway, in this case, the ka in kami is replaced by ga, making it shinigami.
So, to sum up, we have:
Shinu --> shini when the verb becomes a noun
Kami--> gami when stuck on to the end of an I-form-verb.