$\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected?

The answer to your first question is yes: $\{(\leftarrow,0),(0,\to)\}$ is a clopen partition of the irrationals, and its existence shows that they are not connected.

$\Bbb R^2\setminus\Bbb Q^2$, on the other hand, is connected, and even path connected: you can get from any point of it to any other point along a path lying entirely within $\Bbb R^2\setminus\Bbb Q^2$. In fact, you can do it alone straight line segments, using at most three of them; just make sure that each horizontal segment lies on a line $y=a$ with irrational $a$, and each vertical segment lies on a line $x=a$ with irrational $a$. See if you can work out the details for yourself; I’ve written them up below and left them spoiler-protected.

Suppose that $\langle a_,b\rangle$ is a point with at least one irrational coordinate. Without loss of generality let $a$ be irrational. Let $\langle c,d\rangle$ be any other point with at least one coordinate irrational. If $d$ is irrational, you can travel along the line $x=a$ to the point $\langle a,d\rangle$, and then travel along the line $y=d$ to $\langle c,d\rangle$; this path lies entirely in $\Bbb R^2\setminus\Bbb Q^2$. If $c$ is irrational, travel along the line $x=a$ to any $\langle a,u\rangle$ with irrational $u$, then along $y=u$ to the point $\langle c,u\rangle$, and finally along the line $x=c$ to the point $\langle c,d\rangle$; again the path lies entirely in $\Bbb R^2\setminus\Bbb Q^2$.

As the others have noted, the second statement is wrong. One can arrive at a less direct, but maybe more elegant proof of its converse by proving the stronger fact that $\mathbb R^2\setminus A$ is path-connected for every countable set $A$. As far as I remember this simple proof is due to Cantor. I only do not give it here to not spoil anything, I hope that having said the word “countable” is a sufficient hint.

You are right about the disconnectedness of $\Bbb R-\Bbb Q$.

The set $S$, on the other hand, is connected, even path connected. Choose irrational $\alpha,\beta$. Each point $(x,y)$ can be joined to the point $(\alpha,\beta)$ by the path $p:I\to\Bbb R^2-\Bbb Q^2$$$p(t)=\begin{cases} 2t(\alpha,y)+(1-2t)(x,y) &\text{ if }t\le\frac12\\ (2t-1)(\alpha,\beta)+(2-2t)(\alpha,y) &\text{ if }t\ge\frac12 \end{cases}$$ if $y$ is irrational, and $$p(t)=\begin{cases} 2t(x,\beta)+(1-2t)(x,y) &\text{ if }t\le\frac12\\ (2t-1)(\alpha,\beta)+(2-2t)(x,\beta) &\text{ if }t\ge\frac12 \end{cases}$$ if $x$ is irrational