How are integer sets of "all odd positive" or "all even negative" or "all negative" described? The infinite set of all positive integers, i.e., evens and odds combined (sometimes including, sometimes without, $0$), is called the "natural numbers", denoted by $\mathbb{N}$. What about with a specific half of them omitted, or all of them converted to their additive inverse? And does this work the same way for finitely-defined sets?
Slightly off-topic but how might "all complex numbers" or "all rational numbers" (two cases also with special abbreviations) or "all natural numbers" or cetera be described in terms of a subset of "all numbers" (i.e. all numerical values conceivable/possible) without using their respective abbreviations?
 A: 
How are integer sets of "all odd positive" or "all even negative" or "all negative" described?

To answer the question in the title, recall how can we define a set.
We can define a set by extension, listing all its elements (note that this way of defining a set doesn’t contemplate the order and the number of times the elements are listed). This method works fine for finite sets (and “small” sets) since you have to list each element, one by one.
Also, you can define a set by stating one or more conditions that are satisfied by the elements of that set (and only by them). And this is a good approach to accomplish what you want.
If you want the set of all integers that are odd and positive, then you want a set of integers $z$ that satisfies the property of “$z$ is odd and positive”. And you can represent the set as follows
$$\{n \in \mathbb{Z} : n > 0 \wedge \exists k \in \mathbb{Z} : n = 2k + 1\}.$$
Equivalently, you can define the other two sets in you question as
$$\{n \in \mathbb{Z} : z < 0 \wedge \exists k \in \mathbb{Z} : n = 2k\} \text{ and } \{n \in \mathbb{Z} : n < 0\}.$$

What about with a specific half of them omitted, or all of them converted to their additive inverse?

Suppose that you have the set $X = \{1,2,3\}.$ If you want the set of the additive inverses of the elements of $X,$ you can do it as follows (similar to the other three above)
$$\{-x : x \in X\} = \{-1,-2,-3\}.$$

If by finitely-define sets you mean sets described by extension, then the answer is yes as the example above shows.
For the set of all numbers there is no such structure.
But you can describe, for example, the set off all rational numbers and complex numbers without using its abbreviation. Just recall what is a rational number and what is a complex number. By using the notation above you can described this sets as follows
$$\{x : \exists a \in \mathbb{Z}, b \in \mathbb{N} : x = \frac{a}{b}\} \text{ and } \{x : \exists a,b \in \mathbb{R} : x = a + bi\}.$$
