If conditional expectation is a constant, can we infer the independency? If $E[Y\mid X]= c$ where $c$ is the same constant for any $X=x$ (actually it's just $c = EY$), can we say that $X$, $Y$ are independent?

No, otherwise the concept of martingale would not go beyond sums of independent random variables.

Let us give an example: if $Y=\varepsilon X$, where $\varepsilon$ is zero-mean and independent of $X$, we have $\mathbb E\left[Y\mid X\right]=X\mathbb E\left[\varepsilon\right]=0$. In general, $\varepsilon X$ is not independent of $X$.

However, if $\mathbb E\left[e^{itY}\mid X\right]=\mathbb E\left[e^{itY}\right]$ for each real number $t$, then we can deduce that $Y$ is independent of $X$. Indeed, for a fixed $s$, approximate $e^{isX}$ by indicator function of $\sigma(X)$-measurable elements.