1) What is the meaning of "kendi çevresindeki" in the following sentence?
Gezegen şimdi kendi çevresindeki dönüşünü bir dakikada tamamlıyor.
Meaning of it is “around oneself” so the sentence means
“The planet now completed its turn around itself in a minute.”
2) What is the meaning of "dinlenebilmeniz " in the following sentence?
Size dilediğinizde dinlenebilmeniz için bir yol gösterebilirim
Dinlenebilmeniz is a sentence hard to explain it comes from dinlenebilmek - to be able to get rest and dinlenebilmeniz is something like your (in formal usage) ableness to get rest, it sounds weird I know but sentence means
I can show you a way for you to rest when you wish to.
3) What is the meaning of "kaynaklanmasıydı" in the following sentence?
Küçük prensin kendine asıl itiraf edemediği şey üzüntüsünün daha çok bir günde 1440 günbatımı izleyemeyeceğinden kaynaklanmasıydı!
Kaynaklanmasıydı comes from kaynaklanmak which means originate, have its source in and kaynaklanması is something's reason, something's source and last part is the result of a form called “ek fill” it is used to make it the verb. So the sentence can be translated as
The thing that Little Prince couldn't admit himself was that his sadness “actually” had its source in not being able to watch 1440 sunsets in a day.
Daha çok doesn't mean actually but in that sentence it means the reason of something more than other reasons so the only way I could put it was actually.
Hope these help you, good luck.