Addressing the first point:
- Why does Christ actively mediate or advocate after salvation? Does God the Father forget that we're saved?
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:
- Also I've often heard it said that God the Father is against us and Christ is for us, so does that mean that the Father and Son have two different wills? One that hates us and the other that loves us?
- Does God the father love those whom Christ advocates for and is God the father angry at believers every time they sin?
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.