Which of these signals have same Nyquist rate as $x(t)$?
Your understanding is right. 

With the usual Fourier transform pair notation :

$$x(t) \longleftrightarrow X(\omega)$$

and assuming bandwidth of $x(t)$ is $W$, then the four cases will be 


1-) $x^2(t) \longleftrightarrow X(\omega) \star X(\omega) \implies Bandwidth = 2W$
2-) $x(2t) \longleftrightarrow \frac{1}{2}X(\omega/2) ~~~~~ ~\implies Bandwidth = 2W$
3-) $ x'(t) \longleftrightarrow j\omega X(\omega) ~~~~~ ~~~~~\implies Bandwidth = W$
4-) $ x(t)\star x(t) \longleftrightarrow X^2(\omega) ~~~\implies Bandwidth = W$


You can see that, cases 3 and 4 has the same bandwidth as $x(t)$ and therefore their Nyquist sampling frequencies will also be the same.