It means that the Spirit of God works through us. that our work is not about doing what is physical only, it is about the manifestion of the Holy Spirit, (spirit of God) working through us. We must be submissive to God in order for Him to work through us. (We must allow God to use us.)

Here, the Angel is explaining the Holy Spirit and the Will of God, how they work together. This tells us that God works through us.

Jeremiah 10:20 speaks to God's hurt and anguish that His children are stiffnecked and sottish, how they have turned away from Him in disobedience and sin. (So, if we are not attuned to God's will and His instruction, we will not have a mind to do what God wants of us, He will not be able to work through us because we are not opened to His love and His Instruction)

Psalms 51:6 speaks to David being humbled because He felt poor in spirit and destitute, He knew that what He needed, man could not assist him with, nor give to Him. (God had ordained David to teach and instruct, but David erred and realized his disobedience to God, He cried out to God to help Him, He was submitting to God's will, and was seeking forgiveness so that He could be used of God to fulfill his plan for David.) It further speaks to David knowing truth, and what He did not know, He knew that by the gifts of the Holy Spirit, God would give Him knowledge and truth of. that is exercising faith of which we must have to please God. When we are pleasing to God, He can use us and work through us.

So yes, in essence, they can all compliment each other and bring you to the understanding of the Spirit of the Almighty God, How it works, and How it works through us, and what we need to be for God's spirit to work in us according to His perfect plan and will. Everything done to the Glory of God, not for us for we can do nothing of our own and give nothing of our own.

Well, Jeremiah in 10 around verse 20 laments the destruction of the first temple and going to exile in Babylon and Zechariah in chapter 4 and around the verse you speak of lays the capstone of the second temple after the return from the same exile.
So these are most definitely related.

But Psalm 51 is (as the first line tells it) about David asking forgiveness for what he did with Bathshevah. So, lines 5 and 6 contrast the natural proclivities of man with the commandments of God and form the basis for David asking forgiveness for having strayed due to those natural desires.

I suppose you can certainly make the abstract paralel to the rebuilding of the temple through "the Song of Solomon" where the sins of Israel are likened to sexual unfaithfulness -- so the destruction of the first and rebuilding of the second temple are the rejection of the immoral and then forgiveness of the penitent.