1)

Neither data no rvariables are "parametric" or "non-parametric". A distribution model can be parametric, what means that the density, mass or distribution function has parameters that define the shape (or the moments).

There are many skewed distribution families that can be expressed in a parametric form (see e.g. the Handbook of statistical distributions: shorturl.at/cAR47). The log-normal is also a parametric distribution.

2)

The distribution of the variables is not (directly) relevant for the validity of the ANOVA. Concretely, the residuals of the entire model should be approximately normal distributed to get a valid interpretation (that's the conditional distribution). For an ANOVA model this automatically means that the distribution within each group should be approximately normal. But it is usually better to have a look at residual diagnostics plots using the data from all groups together. This generalized to models that include metric predictors (regressions, ANCOVA models, multilevel models).

3)

If using logarithms is meaningful depends on the kind of variables you have and how you want to model effects. If effects are proportional to the expectation (mean), using logarithms is actually recommended. An alternative is a gamma-model with log-link (shorturl.at/bily4, https://stats.stackexchange.com/questions/67547/when-to-use-gamma-glms).

No need to transform the normally distributed variables and parametric tools can be used. On the other hand, nonparametric tools must be used in the analysis of the skewed variables without log transformation. If transformation is conducted, these variables will transformed to normally distributed and parametric Statistics can be used.

Look at these papers

Article Log-transformation and its implications for data analysis

Article Logarithmic transformations in ANOVA