Now, I personally dislike Faith and her insta-redemption
I'm really going to have to disagree with you there. Faith and insta-redemption? From whom? Buffy? Buffy was willing to reason with Faith if she was trying to do better after her coma, but that's because she'd been having these freaky dreams. I don't know about their nature, but I, and Buffy, too, I think, thought she and Faith were sharing those dreams and Faith was a good person in those dreams. But once Faith showed she didn't want to play nice, Buffy was ready and willing to fight.
Or are you meaning redemption from Angel? In Five By Five, before Faith broke down, Angel was ready and willing to kill her. Cordy totally left in Sanctuary because she didn't trust Faith. Wesley helped because he truly believed Faith was trying to change, but he was against it in the beginning. And then Faith turned herself in to the police, to spend several years in prison.
And THEN, when she went back to Sunnydale, Buffy still didn't like her. she didn't trust her, didn't want much to do with her (other than build up a load of sexual tension, that is). Her murdering was still being held against her.
So how is granting Faith redemption? Oh, I know people do. But insta-redemption? I can't think of a single person on the show who instantly, without anything happening, forgave Faith.
As for that whole Ted business, Buffy's "spidey-sense" or, more accurately, intuition, only works if she actually spends time with that person and if that person is bad. In Oz's case, he is neither bad, nor fully a werwolf. When the moon's out and stuff, he is, but he doesn't even remember his wolf times. All but 3 days of the month, he is fully human, I'd say. This compares to Veruca who has embraced the wolf in her. And it's not fair to say Buffy didn't know about Veruca... did the two even have a conversation? The same applies to the guy from the Puppet Show. We never saw him and Buffy having a conversation or even crossing paths. The only time she ever saw Ms. French was in class for all of what? Two days? Ampata, I agree, was a strange situation. But she had been human once, which may account for it. And she wasn't exactly evil. She just wanted to be a normal girl, which Buffy understood.
You're right that Buffy never relied on her intuition to slay anyone, and I don't think that's what she did with Ted. Most humans Buffy hit go down pretty fast. They either pass out or are otherwise immobilized. And with other humans she's beat up, they didn't attack her first. With Ted, he did hit her first AND he remained conscious after Buffy hit him for the first time. So she was both defending herself and using what she believed to be the force needed to do so. That was my whole take on the thing.