Can an extrovert suffer from social anxiety and communication problems? Or is it the "prerogative" of introverts? And then what is the difference between character and problem?

Social phobia does not distinguish between an extrovert and an introvert. Social anxiety is primarily a fear of evaluation and extroverts willingly produce it in the form of fear of "screwing up". For extroverts, their effect is more important, therefore, assessment is extremely important and failure in this is sensitive.

Introverts may find it easier on the one hand, that they do not care more about public opinion and their own presentability, but it is more difficult with communication skills due to for an emphasis on inner experiences. All this can manifest itself in the fear of calling, meeting, bargaining on the market, returning goods, etc.

The most reasonable, as far as I know, the theory of extraversion and introversion is H. Eysenck's theory, which raises these features to brain tone (later studies showed that to the tone of the cortex).

It is assumed that extroverts have lowered it, so they experience comfort and a surge of energy from hyperstimulation, the private and one of the most easily accessible manifestations of which are noisy companies and large crowds.

introverts, it is increased, so they experience all the same from a low level of stimulation, which is easiest to get alone.

In other words, introversion / extraversion most of all characterizes the method of the most effective rest for a person and little more ... And this is an innate, physiologically determined property that does not in any way intersect with what kind of injuries a person will receive in the process of growing up.

In other words, yes, even a bright extrovert can have serious social problems. The difference is that other things being equal, he will be more interested in society than an introvert in a similar life situation.

It is worth starting with the fact that an introvert and an extrovert do little to describe the behavior of a particular person, and rarely any of the specialists conducts diagnostics to identify this fact.

And, as a rule, there are no absolute values ​​on the introversion and extroversion scales. Consequently, an introvert can exhibit introverted traits and vice versa. And nothing prevents the extrovert from suffering from social phobia.