Many of us remember how bad (and ignorant) we were when we first started writing; this is likely the case with your friend, right? Eventually, she'll probably grow up and get her act together, and even if that is not the case, it's not something you should worry too much about. Your friendship is more important.

But should she? Does she really want to post stuff that will make her look like she is clueless?

It would be a lie to say I've never written anything bad. After all, I've done NaNoWriMo and when you're spewing out the words because you're running behind on the 20th quality isn't something you can afford to worry about.

But there are a few things I would say to your friend:
First: Why write a fanfiction if you're not going to stay (reasonably) true to the setting and characters?

I'd point out that if you want people to understand your writing it's important to have decent grammar. It doesn't have to be perfect, but I would make a point of asking for clarification wherever bad grammar makes things confusing. Naturally this includes the paragraph thing.

Sure she can do what she wants. She could sit around on the couch eating twinkies all day, but that doesn't make it a good idea.

but you can't make them drink.

Try what WhiteRaven and Aster have suggested: Be polite and take it one thing at a time.

Sendri is right about if Sam doesn't take the advice well. It's like therapy. The first step is admitting you have a problem. If she doesn't admit her writing has problems, she cannot change it.

Worry about one thing at a time, and only if she asks your opinion.

She asks, "Is it good?" and you say, "I dunno, would (character) really say/do that? It doesn't feel like they're themselves..."

And just leave it at that. Then if they ask you again, point out something different. I know it's hard, but the most luck I have had in creative writing suggestions (was teaching assistant in a kids creative writing workshop) is to point out one thing at a time.

People get discouraged if you dump "These are ALL the reasons why your story sucks" on them. I find the point is not to tell them the story is bad, but point out what parts of it could be better or what parts of it they didn't think through so well...

It's a sad truth that writers are never going to improve until they're ready and willing to do so, and trying to push them only causes hurt feelings. If you want to try, then certainly, be as delicate as WhiteRaven suggests, but this sort of thing often turns nasty, and my advice would be to focus your frienship on aspects of your life other than writing, as much as that's possible.

My sympathies on having to read something to painful :(

Personally, I'd answer her "I can do whatever I want" thing by telling her that if she's any sort of fan, she'll treat the characters and setting with respect and try to do the anime justice. Tactfully point out all the mistakes and ask if she'd tolerate that in a book she was reading, then respond to her "it's only fanfiction" line with what the FAQ has to say on the subject. Try to push her in the right direction with the fic, and if she won't listen, well..

...that's what the PPC is for.

I just skimmed over it, and I have to say, that looks pretty bad. I don't even know the canon, either. Just...the writing itself (everything you mentioned). Wow.

Well, if you're really bothered by it and feel like you should talk to your friend about it, then I'd say to just phrase everything very...delicately. You don't want to ruin a friendship over bad writing, right? Perhaps you can suggest she get a beta or offer to give her some pointers? If she's one of those people who believes that her writing is God's gift to the world, though, it may just be better to leave her be and not bring it up. Spork it if you'd like, or don't. It all comes down to whether or not you want to push the issue with her (and how well would she take it).

Sorry I don't have better advice.