i) $\{(x,y)\in \mathbb{R^2}:2x^2+y^2=1\}$
ii) $\{(x,y)\in \mathbb{R^2}: xy<1\}$
iii) $\{(x,y)\in \mathbb{R^2}: e^x=cosy\}$
iv) $\{(x,y)\in \mathbb{R^2}: 0\le x\le 1, 0\le y \le 1\}$
About the first one: This equation in $\mathbb{R^2}$ is an ellipse with center $(0,0)$, $a=\frac{1}{\sqrt{2}}$ and $b=1$. So it is a closed and bounded subset of $\mathbb{R^2}$. Thus compact. (Also the set is covered by the unit cirle which is compact in $\mathbb{R^2}$)
About the second : I believe this set is a hyperbola but i don't know how to proceed. I don't think it's compact. Maybe you have to make cases about $x$ and $y$. For example if $x\neq 0$ then $y<\frac{1}{x}$ and then examine if the set is compact.
About the third set: $e^x>0$ and $-1\le cosy \le 1$ so we get $0< e^x\le 1$ which is bounded but not closed, thus not compact.
The last set(a square) is closed and bounded in $\mathbb{R^2}$
i) You're right
ii) Both axes are contained in the set, so it is not bounded, so not compact. ($xy=1$ is an hyperbola). In fact, this set is open (not closed).
iii) Any point of the form $(0,\pi/2+2k\pi)$ with $k\in\mathbb Z$ is in your set, so it is not bounded,so not compact.
iv) you're right.