The style of usage with repeated "and" is arguably polysyndeton, as someone else pointed out, however, it is really more to do with the general idiom of Hebrew. Hebrew is a very different language than English, and it would be a mistake to try to project English rules and idioms (such as never start a sentence with "and") onto that language. It is worth remembering, for example, that early Hebrew texts did not have any punctuation, so the idea of "sentence" is rather different in early Hebrew.
A case might be made that the translators should have rendered the Hebrew idiom better into English idiom, as is done in many modern translations, but that is a subtle question about what "translation" means, and is way out of scope for this discussion forum.
As to whether using this polysyndeton structure in English conveys an archaic meaning, personally I don't think so. There are many other features of the KJV that do this better, such as the use of "thee" and its friends, the more diverse verb conjugations, and the unusual sentence structures. The repeated use of "and" — not so much.
I can comment more extensively on "LORD". This is a translation of the Hebrew word commonly transliterated as either Jehovah or Yahweh. In fancy circles they call it the tetragrammaton — which simply means "four letters," but is long and intimidating enough to keep out the riff raff. It is a four letter word used, arguably, as the personal name of God throughout the Bible. In the traditional reading of the Hebrew Bible this word was always considered too holy and pure to actually pronounce, and so they used a technique called kitiv-qere. This means "written/read", and essentially means that the word is written one way but pronounced a completely different way. In this case, the word is pronounced not as Yahweh but as Adonai, which is the Hebrew word for "Lord."
This tradition is also projected into the Greek of the New Testament and a Greek version of the Old Testament called the Septuagint which is commonly quoted in the New Testament.
When the English translators translated the Bible into English they wanted to respect this tradition so they translated it as it would have been read as the word "Lord", but to indicate that it was Yahweh and not Adonai which is to say that it referred to as the name of God, they distinguished it by capitalizing it.
There are some places where the grammar doesn't allow this, usually where Yahweh is preceded by Adonai. Translating The Lord LORD God..." seemed a bit redundant. So in these cases the kitiv/qere was to pronounce it as "Elohim", the Hebrew word for God. In these cases, you will see it translated as GOD in capital letters, usually it is Lord GOD.
There is a lot more to this than would be appropriate to convey in this answer. If you are interested in this, you should read the translators introduction to any modern Bible which will tell you more than you wanted to know. Alternatively, check out Wikipedia on tetragrammaton for a great article on the subject.
The difference in the first chapter of Genesis reflects the underlying Hebrew fairly accurately. There is a theory as to why they are used differently. It is called the JEDP theory, or "The Documentary Hypothesis." Personally I think it is bunk, but Google and Wikipedia will tell you all you want to know about it.