So I started reading the Proj construction. I wanted to get more understanding by consider the graded ring
$$\frac{R[x,y]}{(x^2)}$$
where $x,y$ have degree $1$. Let $X= Proj(R[x,y]/(x^2))$. So I want to know if
(i) The scheme is not reduced.
(ii) The scheme is not affine.
For (i) I believe it is not so, since $$O_X(D_+(y))= (R[x,y]/(x^2))_{(y)} \simeq R[a]/(a^2) $$ is not reduced ring. But how does one show that is not affine?
This scheme is not reduced, but it is affine (more or less by accident - more variables would give you a projective and not affine scheme). This scheme is $V(x^2)\subset \Bbb P^1_R$, which is a fat $R$-point, and which has as it's reduction $V(x)\subset \Bbb P^1_R$, just an $R$-point, or $\operatorname{Spec} R$. As a scheme who's reduction is affine is again affine, $V(x^2)\subset \Bbb P^1_R$ is also an affine scheme - in particular, it's $\operatorname{Spec} R[x]/(x^2)$.
If you increased the number of variables, then you would not end up with an affine scheme - for example, $\operatorname{Proj} R[x,y,z]/(x^2)$ is a nonreduced $\Bbb P^1_R$ which is not affine. Dimension zero projective schemes are also affine (and thus finite), and correspond to $\operatorname{Proj}$ of dimension-one rings, like the one you've used here.