From the linked table, it looks like your model isn't "significantly different" from a null model. As you are new to statistics and your questions would be important with "significant" results, here are some answers.
Because the design is unbalanced, I subsequently fitted a Type II ANOVA model with the same two main effects and their interaction. Is this procedure appropriate?
With an unbalanced design, results with Type I ANOVA depend on the order in which the predictors are entered into the calculation. See this page and its links. I would argue that it's generally inappropriate to use Type I ANOVA with unbalanced data. Your use of Type II ANOVA with your unbalanced data is certainly appropriate.
One caution: don't let p-values/significance of prior calculations determine later choices of tests. In your situation, your choice of Type II ANOVA could be made without having done the prior Type I ANOVA, based on the unbalanced design.
Due to a significant Shapiro–Wilk test (I am aware that this test has limitations for small sample sizes, but it was required by my supervisor), I applied a squared transformation to the dependent variable.
There's nothing wrong in principle with transforming dependent variables. If you have reason to believe that the predictors are linearly associated with the square (or the log, or the square-root, or...) of the dependent variable, that's fine. Just recognize that you are then modeling the mean of the square (or of the log, or of the square-root, or ...) instead of the mean of the variable itself.
As you learn more about statistics, look into other types of models that might be more directly related to the questions you are asking. For example, generalized linear models can be more useful than simple transformation of dependent variables. Ordinal regression doesn't make assumptions about distributions of outcomes or error terms, and with modern computational tools it can now be applied in practice in many situations that would previously have been analyzed with ANOVA or multiple linear regression.
My supervisor suggested reporting an effect size. Which effect size measure would be appropriate in this case, and do I need to back-transform the variable before calculating it?
The proportion of variance explained by each factor is the basis for frequently used measures of "effect size" in ANOVA. This page explains how to use tools in the R effectsize package to perform several types of such calculations. You have to be careful here with an unbalanced design; make sure to calculate in a way that uses your Type II ANOVA result, as explained on that page.
As you are modeling the square of the dependent variable, the "proportion of variance explained" from your Type II ANOVA will be with respect to the variance among the squared values. That's OK to report if you are clear about that. It just might not be as intuitively easy to understand as if you had performed ANOVA on the original outcome scale.
As you learn more about statistics, be very careful with the term "effect size." It has many different meanings. When you see someone use that term (or you use it yourself), make sure that you know just which "effect size" is in question.
