As I understand, both "an" egg and "the" egg are said the same way in Russian, so either would be valid
Why isn't it necessary to include меня to convey the meaning of give ME an egg, as opposed to just give an egg?
You can include «мне» in Russian. But it should be «мне» (Dative case), not «меня» (Accusative case; here, «яйцо» is this case).
I'm not sure why this exact translation has «me» in English but doesn't have it in Russian. Maybe it would sound unnatural in English without «me»?
Could you pass the salt to me, please? Could you pass me the salt, please? Could you pass the salt, please? Pass the salt, please. Salt, please.
All sound perfectly okay. If it were me speaking, I would use any of the shortest 3 variants for brevity. The longer ones carry a tone of being more polite. But they all still sound perfectly fine.
"Pass the salt". "Me" is not always necessary... (BTW, I haven't tried using "pass" instead of "give" for дай/дайте, so I don't know if would be accepted. It should be, for this context)
So the direct translation is something like "give eggs, please"? Is it always naturally inplied that they need to be given to you?
Maybe it is similar to the way Korean speak. We casually say "pass or give the egg" when it's meant giving to me.