Intercession is a general term, used even of the Holy Ghost in us (Rom. 8); but priesthood (in Hebrews) is with God, for mercy and grace to help in time of need: advocacy with the Father, to restore communion when we have sinned You could not have it for sins in Hebrews, because the worshippers once purged have no more conscience of sins. This answers your three first questions save the end of the third; Why do we fall? Because it is part of the government of God to have us responsibly exercised, though not without grace sufficient for us and strength made perfect in weakness. But if we forget our weakness and dependence, we forget the grace too, and are in the way of a fall: see Peter's case, and the Lord did not ask he might not be sifted; he wanted it. The evil is not in the fall, really grievous as that is, but in the state it manifests. God may allow it that we may learn this.
Washing the feet is in connection with the advocacy—we have dirtied them. "Save," in Heb. 7:25,25Wherefore he is able also to save them to the uttermost that come unto God by him, seeing he ever liveth to make intercession for them. (Hebrews 7:25) is securing across the difficulties and dangers on to the end, as "if the righteous scarcely [μόλις, with difficulty, across what brings ruin if [we are] not kept, as Noah, Lot] be saved."
"Able to succor," as in chapter 2:18, refers not to strength, though of course it must be there, but experimental knowledge of the opposition, difficulties, trials, which are on the road, so that He could understand, be touched with them. The priest does represent us, "appear in the presence of God for us," but that is before God, but He also obtains for us all needed grace and help, as regards the way down here. And learning our dependence, and to trust in God's faithfulness is a great thing; man would be independent, and has to learn his relationship to God, or rather know himself and God in it. This has its importance as well as being perfectly accepted.
Ever affectionately yours in the Lord.
July, 1874.