The more insight one gains by meditation and studying the suttas, does one also gain a higher proficiency in teaching the dhamma?
You could be misunderstanding or misinterpreting your meditative experiences. So meditation on its own does not make you fit to teach. (Many wrong views in Brahmajala Sutta arise due to meditative experiences.) You should have theoretical knowledge also and validating the experience with the teaching. Also you should have good communication skills.
To put up an extreme example, if someone attains nirvana/full enlightenment, is he as proficient/competent in teaching the dhamma as the Buddha himself was?
Dūta Sutta discusses qualities of a good messenger or teacher of some which I glossed over above. Nobody will have skills in teaching compared to the Buddha but if you were to aspire to become skilled what you should cultivate are the analytical skills related to teaching:
- connotative meaning
- denotative meaning
- verbal expression
- analytic insight
Mahā Vedalla Sutta, (Saṁyojana) Kotthita Sutta
Also see: THE TEACHING METHODS OF BUDDHA by Piya Tan
I read that there can be Buddhas, which attain enlightenment by themselves, but neither have the fellowship nor the necessary skill to teach the dhamma, is this true?
Yes I believe to to be true. They are Paccekabuddhas.