Does the set $left{ left(begin{array}{c} x\ y end{array}right)inmathbb{R}^{2}|xleq yright} $ span all of $mathbb{R}^2$

The usage of "group" word is misleading here, especially with the question tagged group theory as well.

So let consider the vector space only question. $mathbb R^2$ needs only two independent vectors to be spanned.

It is clear that $(0,1)$ and $(1,2)$ are independent and both belong to your initial set, so these two vectors just by themselves span $mathbb R^2$. A fortiori the whole set also does.

I could have taken $(1,1)$ as in azif00's answer, but this is just to show that even with $x<y$, the question remains mostly unaffected.

I don't understand your argument, but $dbinom 11$ and $dbinom 01$ are elements of $bigg{ dbinom uv in mathbb R^2 : u leq v bigg}$ such that $$binom xy = x dbinom 11 + (y-x) dbinom 01$$ for every $dbinom xy in mathbb R^2$.

Let $V$ be the span of the specified group. Fix an arbitrary $(x,y) in mathbb{R}^2$. Ix $xleq y$ then clearly $(x,y) in V$. If $x>y$ then $-x<-y$, so $(-x,-y) = -1(x,y) in V$. So yes, it spans.