Autoconvolution vs Autocorrelation
Are two signals are the same if their auto-convolution functions are the same?


Almost. Look at the autoconvolution in the frequency domain where the autoconvolution of $x$ (with itself) gives us $(X(f))^2$ in the frequency domain while the autoconvolution of $-x$ (with itself) gives us $(-X(f))^2 = (X(f))^2$. So, given an autoconvolution function, there are two (very related) signals $x$ and $-x$ that have the same autoconvolution function.


  Are two signals are the same if their auto-correlation functions are the same?


Not quite.  Now we are given $|X(f)|^2$ in the frequency domain and there are many different factorizations possible. For example, if $y(t)$ is a signal such that values taken in by its Fourier transform always lie on the unit circle in the complex plane (for every $f$, $|Y(f)|=1$) then $|X(f)Y(f)|^2 =  |X(f)|^2$ and so $x\star y$ has the same auto-correlation function as $x$. Note that $x(t)$ and $x(t-\tau)$ (which is a delayed version of $x(t)$) have the same autocorrelation function ($Y(f)$ happens to be $\exp(-j2\pi f \tau)$ here). Another factorization replaces $X(f)$ by $X^*(f)$ which tells us that $x(t)$ and $x(-t)$ (which is just $x(t)$ running backwards in time) have the same autocorrelation function.