This is a messianic vision, and David is primarily concerned here with Christ and his place with God, and his Priesthood Authority.
I see the first part of your question as having two elements. Element 1, "The Lord says to My Lord" is a conversation between God The Father (Elohim) and Christ The Son (Jesus Christ). Element 2, it was necessary to distinguish a conversation between two who could both be described as God, or Lord. The reason one is "The Lord" and the other "My Lord" is because though God The Father is creator of all, David knew that there would be a more direct relationship with The Christ as a personal savior and redeemer of his house, and Israel as a whole. Hence Christ being David's Lord, or "My Lord."
As for the NT portion of your question, I can point to Matthew 22:41-46, as a clearer reference that this meaning as I explained was understood NT times, Jesus Christ was the Lord of Psalm 110, of David, and of the House of Israel, and "The Lord" was God The Father.
41 ¶While the Pharisees were gathered together, Jesus asked them,
42 Saying, What think ye of Christ? whose son is he? They say unto him, The Son of David.
43 He saith unto them, How then doth David in spirit call him Lord, saying,
44 The Lord said unto my Lord, Sit thou on my right hand, till I make thine enemies thy footstool?
45 If David then call him Lord, how is he his son?
46 And no man was able to answer him a word, neither durst any man from that day forth ask him any more questions.