whats the official name of the earth and the moon?

Mike and Moony.

Sol is the name for our son, and we're the 3rd planet out from Sol in our "Sol"-ar system, so you could call our planet Sol 3 but it's actual name is "Terra". We call our moon, "Luna".

The names of the earth and moon are Earth and Moon.  The earth is 'the earth',  uncapitalized, unless you're talking about ALL the planets, and then it's Mercury, Venus, Earth, Mars, etc.

Sol is the Latin name for the sun. The Latin name for the moon is Luna.  But I don't think those names are used in science except as adjectives--solar and lunar. Sol 3 means the third planet from the sun, and that's how we'd refer to the earth if the solar system was newly discovered and the planets didn't have names.  Just like nobody would address a letter to a house as "The third house on the left after you cross Oak St.", though you might do that if the neighborhood had just been built and there were no house numbers.

Earth, and the Moon. Since there is only one planet Earth, it is known simply as "the Earth."

"Terra" is not a correct name for Earth, nor are "Sol" and "Luna." These are simply the Latin for Earth, Sun, and Moon.