I’ll answer you questions the other way around:
Are PCA and factor analysis the same thing?
According to some definitions PCA is a type of factor analysis (where the term is used to describe a general family of methods such as here http://www.statisticssolutions.com/factor-analysis-sem-factor-analysis/ which explicitly includes PCA as a factoring method and here with a definition that allows PCA http://www.statisticshowto.com/factor-analysis/). According to others a much more strict as specific definition lead them to conclude they are separate (thanks to @Richard Hardy for the useful links in the comments that explain this in some depth). Factor analysis basically a set of tools to uncover hidden patterns in your data, which is why some people broaden the definition to include methods such as PCA which has the same basic aim. PCA specifically seeks to maximise the covariance (or correlation when appropriately scaled) in as few orthogonal factors as possible. Other methods can impose different limits (non-zero, orthogonality in scores as well and many more).
How do I estimate the relationship between the extraversion dimension and problematic behavior?
PCA will not return pure relationships between specific variables and an issue you are interested in. There are many different tools that are designed to do this (by the sound of it discriminant analysis may be relevant if you have categorical (problematic behaviour and non-problematic behaviour).
However, PCA will be a powerful tool to explore your data. What PCA factors will reveal is how the extraversion behaviour interacts with the other dimensions. It will reveal what variables it tends to coincide with, which ones it is unrelated to and which ones it is inversely related to. The first PCA will show you the largest source of variation and so the strongest relationships between the different variables. Each PC after that describes an ever decreasing proportion of the variation, revealing different sets of interactions. Typically in real-world datasets such as yours, the hope is that some of these natural groupings of interactions is associated with some independent variable (in your case problematic behaviour). You would examine eigenvalues and restrict your focus to those >1 (this means the factors that describe at least the same amount of information as the original variables, other constraints can be used if this still leaves too many PCs to go through). Carry out statistical tests on the scores (scores are basically the proportion of those PC factors in each sample) to see if any of the factors provide a significant difference between your groups.
If none of the PCs naturally describe a difference then discriminant methods are worth looking at, but as they deliberately reduce the data to enhance contrast in your grouping variable one must take great care to validate the model to ensure it is not overfitting.
If you specifically want to know how one dimension interacts with your dependent variable then use the relevant univariate test. The advantage of PCA compared to sequential independent statistical tests on each variable is that it summarises how each interacts and gives a more complete picture. So you will understand which other dimensions need to coincide with extraversion to make subjects more likely to be problematic.