Is the phrase "The Lord Be With You" and the response "And with your spirit" directly pulled from the Bible or where did that come from?
It is directly pulled from the Bible.
Based on my answer to this question: As far as scripture goes, the salutation is from Ruth 2:4 and 2 Chronicles 15:2 in the Vulgate.
In Ruth, the phrase appears in the sentence, "Et ecce ipse veniebat de Bethlehem dixitque messoribus: 'Dominus vobiscum'. Qui responderunt ei: 'Benedicat tibi Dominus'." ("[Boaz himself] came from Bethlehem and said to the harvesters, 'The Lord be with you!' and they replied, 'The Lord bless you!'").
II Chronicles recounts that Azariah said, "Audite me, Asa et omnis Iuda et Beniamin! Dominus vobiscum, quia fuistis cum eo. Si quaesieritis eum, invenietur a vobis; si autem dereliqueritis eum, derelinquet vos." ("Hear me, Asa and all Judah and Benjamin! The LORD is with you when you are with him, and if you seek him he will be present to you; but if you abandon him, he will abandon you.")
The phrase additionally appears in Numbers 14:42: "Nolite ascendere: non enim est Dominus vobiscum: ne corruatis coram inimicis vestris.”
St. Paul words to Timothy are alluded to: "The Lord [Dóminus] Jesus Christ be with thy spirit [cum spíritu tuo]. Grace be with you [vobíscum]. Amen." (2 Tim. 4:22).
By "you" and "your spirit" are not meant distinction between spirit and person, but goes back to the Hebrew notion that you can speak of someone by referring to their soul or spirit, which are often conflated in Hebrew thought; though this doesn't imply no distinction existed. E.g. Ps. 3:1-2: "How my enemies are multiplied; they abound who rise up against me. They say to my soul, 'There is no salvation for him in his God.'"