1. Here is my approach to it. Usually we have set of integers modulo or residue classes mod . Either way this set contains all the integers mod , which means this set consists of the remainders after division by of the integers. And we are choosing the smallest positive ones.
For we are dealing with infinity, so it can be tricky, but lets try to do the same thing. Infinity is usually defined as being something that is greater than any positive integer, that is, if , then . So for any positive integer we already have that is smaller than the thing we are modding out by, so we have that is its own remainder. Thus
Now, what about the negative integers. It gets trickier For a positive integer we already have . Multiplying by we get . We need some help from to understand what is going on. When we have , we have , so when we mod by , the least positive remainder is . If we try to apply this to the the case with infinity, we must get that the least positive remainder is . So then we have non-negative integers.
Now don't really take my word for it. It's just how I would think about it. If your book has not mentioned what means then it probably does not matter that much what you do as long as it's logical and coherent. If it's actually important in the framework of the book, then there must surely be a definition of what means somewhere in the book. If you haven't checked thoroughly, check now.
2. In this question you are asked specifically about so I think you really don't need to worry about . Among the integers, there are no zero divisors since for any non-zero , you have . Now, lets talk about the units. For an element to be a unit, it must be invertible, that is there must exist an element such that . Since we are talking about the integers, the only integers with this property are clearly 1 and -1 since if we consider anything greater than 1 in magnitude, we have that the product has magnitude greater than 1.