What is a distribution-wise asymmetric measure?
What is the meaning of distribution-wise asymmetric measure?

The (forward) KL-divergence is distribution-wise asymmetric because if you calculate it as $$D_{KL} \langle P(X) \Vert P(Y) \rangle$$ where $P(X)$ and $P(Y)$ are two different probability distributions with the latter being the reference distribution, then $$D_{KL} \langle P(Y)\Vert P(X)\rangle \neq D_{KL}\langle P(X)\Vert P(Y)\rangle.$$ In other words, the reverse KL-divergence is not equal to the forward KLD. If forward KLD were symmetric, then the above would be an equality, not an inequality.

Is there a symmetric measure?

A distribution-wise symmetric measure would, for example, be mutual information:
$$I(X;Y) = H(X)+H(Y)-H(X,Y) = D_{KL} \langle P(X,Y) || P(X) \cdot P(Y) \rangle,$$
where $H(X)$ is the entropy of a variable $X$'s probability distribution, since $I(Y;X) = I(X;Y)$. Mutual information is a special case of the KLD in which the joint distribution is measured against the product of marginal distributions.

What are the rules that a quantity should follow to be qualified as a statistical metric of spread?

The three axioms that a distance metric should meet are:

Identity of indiscernibles
Symmetry
Sub-additivity or triangle inequality

Since mutual information does not obey the triangle of inequality, it does not fit the full criteria for being a distance metric. Instead, variation of information does meet all of the above requirements and is a true metric:
$$VI(X;Y) = H(X,Y) - I(X;Y)$$
where $H(X,Y)$ is the joint entropy.