In all my life, the word scheme has always held a negative connotation, typically involving criminal acts or fraudulent acts.
That's exactly why nobody who truly meant it would ever describe what they were doing as a scheme. It's just not how criminals talk.
We have two people who came up with a premeditated plan
Accusing the President of the United States of rape is a serious matter, and one that I'd expect to see some premeditation over.
to hurt Trump politically because they didn't like his politics
The excerpt you quoted says nothing about them being motivated by their dislike of Trump's politics. They certainly wanted to hurt Trump politically - because of the rape.
The friend is also an alibi who verified that she was told about the rape at the time it happened.
Presumably that's the whole reason Carroll contacted her in the first place, because she was there and she knew about it. There's no suspicious "also" here that needs an explanation. You're basically pointing to someone already involved in the case and saying "What are the odds that this person involved in the case...would turn out to be involved in the case?"
Jean Carroll did not scream when it happened.
Many rape victims don't. Trauma and paralysis often take hold during such an encounter, as well as the fear that their rapist will retaliate against them if they scream or resist.
She did not tell the police.
Many rape victims don't. They often fear that they won't be believed by the police, or feel ashamed that they ever "let" it happen to them.
She did not write about it in her ongoing diary that she was keeping.
Again, shame and embarrassment can lead to rape victims trying to "omit" the incident by pretending it never happened, which would lead to them not mentioning it in a diary. Not that you'd even believe it happened if she had written about it in her diary.
The first we hear she started speaking about it is in a book she wrote shortly after plotting with her friend on a scheme to get Trump.
Yes, she wrote her book and sued Trump after he was elected. I repeat, this line of argument only sounds suspicious if you don't take the time to think about it. Of course a rape victim who had up to that moment kept quiet could be compelled to speak out after their rapist had become the most powerful person in the world. Of course a rape victim could be more invested in stopping their rapist from being the most powerful person in the world than in seeing their rapist as a private citizen be punished for their crime.
And yes, she discussed and made plans with her friend who partially corroborated her story before she took the momentous step of accusing the President of the United States of raping her.
A jury, too, also assessed this and rejected the claim that she was raped.
I don't know how you can in good faith keep repeating this point while completely ignoring the rest of the story. Carroll said she was raped, and the jury disagreed and said she was sexually abused. Is this a significant repudiation of her story? I would say
no, but setting that subjective point aside, we then have Trump's side of the story. Trump said that nothing between him and Carroll ever happened. The jury disagreed and said that not only had Trump sexually abused Carroll, but that he had been lying and defaming her as a liar when he denied the incident happened. Who comes out of this exchange looking better? The woman whose charge of rape was downgraded to sexual abuse, or the man whose claim of complete innocence was downgraded to being found liable for sexual abuse and defamation?
In the end we are supposed to believe that in a 1996 department store a 50 year old billionaire named Donald Trump, who could and did get models much younger than himself, could not resist forcing himself upon a 52 year old liberal sex advice columnist named E. Jean Carroll. 
Men of all ages, appearances, and occupations have committed rape, and women of all ages, appearances, and occupations have been the victim of rape. Rape is an act of power, not passion. That being said, though, it was amusing during the trial when Trump first insisted that Carroll wasn't his type, mixed up Carroll with his ex-wife Marla Maples, and was forced to admit that Maples (and presumably Carroll too, by extension) was in fact his type.