There are plenty of questions out there asking what complex numbers mean and I never seem to get any of them.
I have a few specific questions i want to ask about complex numbers.
1) what is the difference between $ (x,y) \in R^{2}$ and $(x,y) \in C$?
2) would something like $\hat{x} + \hat{y} + i$, where $\hat{x}$ and $\hat{y}$ are unit vectors and i is $\sqrt{-1}$, make sense ?
As vector spaces, $\Bbb R^2$ is the same as $\Bbb C$. $\Bbb C$ also has a multiplication operation whereas we don't think of $\Bbb R^2$ as having one.
Regarding your first question: The first object is an ordered pair of real values $x$ and $y$. The second object is a single complex value, represented as an ordered pair and denoting the value $x+iy$. One might say, informally, that the second is a "use" of the first. In the context of that use, one might say things that would not make sense generally with ordered pairs; for instance, one might write
$$ (3, 5)^2 = (-16, 30) $$
since $(3+5i)^2 = 3^2-5^2+2\cdot3\cdot5i = -16+30i$, although the ordered pair notation would probably require some clarification before you could simply trot that out.
I'm not sure what your second question is really driving at: make sense in what kind of context?
This has probably been answered before on the site, but anyways
1) Complex numbers can be considered as, and are usually defined as, ordered pairs of real numbers. Hence, every complex number can be considered an element of $\mathbb R^2$. But the connotation is different. When we say $z \in \mathbb C$ we are referring to $\mathbb C$ as the set of all ordered pairs of real numbers along with a special definition of multiplication and addition of numbers.
For 2), there are uncountably many unit vectors, and you can associate with each one a complex number with modulus one. If you do that, you can add a vector with a complex number. But this is unusual notation and is confusing, so I don't recommend it.
