"Physical" meaning of higher moments (their values and their existence)

For your final statement the classic distribution to consider is the log-normal distribution. It is an example of a case where all moments are finite, but the moment generating function does not exist and the moments do not determine the distribution.

Take for example the log-normal distribution with parameters $\mu=0$ and $\sigma = 1/\sqrt2$ which has a distribution on positive values with density $$f(x)=\frac{1}{ x^{1+\log_e x}\sqrt{\pi} }$$

This has finite moments about $0$ of $E[X^n]= \exp(n^2/4)$ but the density of the tail does not decay exponentially.

There is actually a quite simple visual interpretation to all the higher central moments.

To ease the interpretation, and without loss of generality, assume that the moments refer to centered and standardized random variables.

Let $Z = (X-\mu)/\sigma$ and $V = Z^k$, and let $p_k(v)$ denote the probability density or mass function of $V$. For odd $k$, $p_k(v)$ has positive and negative support; only non-negative support for even $k$.

Here is the visual: The $k$th central (standardized) moment of $X$ is equal to the point of balance of $p_k(v)$.

Since the transformation greatly dilates values where $|Z| >1$ and contracts values where $|Z| < 1$, the point of balance is mostly determined by the extremes where $|Z| > 1$.

In the case of odd $k$, this point of balance is determined by the relative extremity of the left and right tails of the distribution of $X$ in the portions of the tails that are most amplified by the given power $k$. Higher $k$ amplifies more extreme portions of the tails.

In the case of even $k$, this point of balance is determined by the overall extremity the tails of the distribution of $X$, without regard to whether left or right, again in the portion of the tail that is most amplified by the given power $k$. Again, higher $k$ amplifies more extreme portions of the tail.

It is worth noting that this visual representation shows that the appearance of the center of the distribution of $X$ is all but irrelevant. In particular, "peakedness" or "flatness" interpretations of even moments are erroneous.