Is there a term analogous to factoring, for addition? e.g. $12$ can be expressed as a product of its factors: $2 \times 6, 3 \times 4$. The composites can be decomposed further.
$12$ can also be expressed as a sum of its somethings: $1+11, 2+10, 3+9, ...$. The non-$1$'s can be decomposed further.
It seems to me that if primes are the atoms of multiplication, then $1$ is the atom of addition. (BTW and maybe the Fundamental Theorem of Arithmetic should be called the Fundamental Theorem of Multiplication)
 A: You are absolutely right, including your BTW. Actually, the notion of prime and irreducible can be defined in any commutative monoid. 
The two cases you consider correspond to free commutative monoids.
Details. The monoid $(\mathbb{N}-\{0\}, \times)$ is a free commutative monoid over the basis consisting of the prime numbers. That is, every positive integer has a unique decomposition as a product of prime numbers. 
Similarly, $(\mathbb{N}, +)$ is a free monoid with basis $\{1\}$, the only prime element. Still with addition, $(\mathbb{N}^k, +)$ is a free commutative monoid with a basis consisting of $k$ primes. For instance, the basis of $(\mathbb{N}^3, +)$ is $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$. 
A: Regarding the terminology
Wikipedia states:

The numbers or the objects to be added in general addition are collectively referred to as the terms, the addends or the summands;

Apparently, one can even use the designation augend to distinguish between 1st and subsequent summand ($\underbrace{1}_{\text{augend}}+\underbrace{11}_{\text{addend}}$).

Some authors call the first addend the augend.

In your example:

$12$ can also be expressed as a sum of its terms/addends/summands: $1+11,2+10,3+9,\dots$. The non-$1$'s can be decomposed further.

