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Addressing the first point:
I'm not sure what you mean by interceding or advocating after salvation, perhaps you misunderstood that the fact that he intercedes and advocates for us is what brings us salvation; we are only saved because he intercedes and advocates for us. Or, perhaps you just meant "after" to imply pursuance, and your focus was just on why he "intercedes and advocates" which your next two points denote.
Addressing points 2-3:
The Father cannot stand wickedness and wrongdoing.
Psalm 11:5
The Lord tests the righteous, but his soul hates the wicked and the one who loves violence.Isaiah 13:11
Thus I will punish the world for its evil And the wicked for their iniquity; I will also put an end to the arrogance of the proud And abase the haughtiness of the ruthless.
Yes, God hates those who do wrong, and must punish wrongdoing, he thirsts to right what is made wrong, and all of us have done wrong to such an extent that we do not comprehend. So yes, he is angered by us, and he will only turn away his anger if we turn to his Son. And if, on top of our sins, we reject his Son, what do you think should be done?
Romans 6:23
For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord.
The Father loves us (and love is not just a stronger form of like, but means to truly, selflessly care about what is best for someone), but he does turn to hate those who know him and learn of all his goodness, and then still choose wickedness; those who fully give in to the selfish, destructive way of life, and have no sorrow or comprehension of what they do to others, that is what evil is.
And if you would like more references, here are some good ones about him hating sin, and here are some about him punishing the wicked.
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The OP asks
"Why does Jesus intercede or Advocate after salvation?"
Because Jesus is the way through whom all comes to be in the new kosmos - the Kingdom. He is an intermediary from humanity to the Father. We are saved to the extent we abide in Christ. To the extent we abide in Christ, we stop sinning.
1 John 2:1
"My little children, I am writing these things to you so that you will not sin. But if anyone does sin, we have an advocate before the FatherβJesus Christ, the Righteous One. 2 He Himself is the atoning sacrifice for our sins, and not only for ours but also for the sins of the whole world."
John then continues at 1 John 3:5
"But you know that Christ appeared to take away sins, and in Him there is no sin. 6 No one who remains in Him keeps on sinning. No one who continues to sin has seen Him or known Him."
The OP asks
Why does Christ actively mediate or advocate after salvation? Does God the Father forget that we're saved?
One is saved because one abides in Christ.