Everything we discover in the sky get eventually a code name, like NGC 7293, Simeis 147, etc.
Does Earth/Moon have a code name too? Or it is just Earth/Moon, etc.?
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Those code names all come from certain catalog. For example, NGC means 'New General Catalogue'. There are various catalogs aiming at different objects, like stars, nebulae, galaxies, etc, but not for the Earth, at least not yet. You can find almost all known astronomical catalogs and tables at CDS
Those aren't "CodeNames": they are simply catalog numbers in widely used catalogs of astronomical objects, such as J. L. E. Dreyer's New General Catalog published in 1888. This was the last and most complete catalog published before astrophotography became widespread, so is frequently used by amateur observers because it includes virtually every object visible with amateur telescopes.
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. You will never see astronomers using these terms, except perhaps in a jocular manner. The only people who use them are science fiction writers.
While "Earth" is a poor name, meaning that it is the same meaning/use as "dirt", and it forces us to use names like "regolith" "Aerology" and other words instead of earth, it is not as unspecific and unscientific as "Sun" and "Moon". Star is a pretty good term, but then we have to say "X Planet's Star" instead of "X Planet's Sun". Makes for difficulty. Moon is a term for a class of satellite. Our moon is called "Luna", or Celene, or any number of other names depending on your language. Science uses Latin, so Luna is a good choice. Astronomers and other scientists DO use Sol, Terra, Luna, Ares, and others. If they didn't then they would say "Sunar eclipse" instead of "Solar Eclipse", "Moonar Landing" instead of Lunar Landing, and my favorite "ExtraEarthestrial". Yes, Sol-3 and Terra are very valid scientific names for Earth by whatever bar you choose, and no matter what any catalog says. They are valid prima facie. So is the symbol of a circle with a cross inside (UTF-8 code 0xC5 which for some strange reason is not supported by this form's useable font...Admin, how about adding Symbol font and math notation stuff to a "smiley" set?)
Do "Sol III" and "Sol IIIa" count as code names?
Since we don't live on other planets, we haven't needed a unique name for the place we live. The best inspirations for names will probably come from science fiction.
Historically (1450-ish or so) the traditonally consented name in most western cultures is the Roman word Sol. "The Sun" is also considered proper nomenclature. Thus, it is likely that "Earth" (also considered proper,) as the third satellite of Sol is either properly Sol III, or Sol-3. Our Moon (again, considered proper) would be Sol IIIa, or Sol-3a. Also correct would be Sun III, Sun-3, Sun IIIa, and Sun-3a. As a side note, common science fiction convention would mean that we are not in fact Humans or Terrans but Solans or Sunnans.
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