"Let me read this message!"
Translation:Дай мне прочитать это сообщение!
Is "мне" necessary in this sentence, or would "Дай прочитать это сообщение" be sufficiently clear?
"Пусть" can only be used with the third person. "Пусть он прочитает это сообщение" (Let him read this message) would be correct.
Still wrong. I don't see how it could be translated with пусть, it should be with a verb in imperative mood, and пусть is a particle.
Yes, this is common. "Let" is just the second meaning of "дать", so "дай мне" is, in fact, the best translation of "let me."
I guess they can be used, but they both are closer to "allow me to" with "позволь" having a bit of old fashion tint to it.