Why was Amy not arrested?

First up, just to get it out of way, the FBI was never involved. The detectives were probably from the local PD. Now, back to the question of implicating Amy for the murder of Desi.

Even though Amy came clean with Nick and told him about killing Desi, there just wasn't any evidence for an investigation against Amy. The CCTV footage from Desi's lake house, the bruising that she gave herself, all were consistent with rape, abuse and unlawful confinement. She had brilliantly orchestrated and disguised the murder as her self-defence and escape, making the entire thing more or less an air-tight case in her favour. And whatever little doubt that people like Detective Boney had, was swept under the tide of emotions surrounding the abduction and subsequent homecoming of Amy. So much so that when Detective Boney tried to prod her on the weak links in her story, she was promptly snubbed by her senior.

There was no opening to force an investigation on Amy. And yes, there was no justice for Nick or Desi.

The police not looking into the murder of Desi whatsoever actually made no sense whatsoever. I understand there were emotions, and KeyBrd's answer makes sense to a degree, but it completely falls apart due to the fact that tape footage does not add up to her orchestrated story at all. Like not even a little bit.

1. The footage of her entering Desi's house was WAY after her disappearance. 5 days, 2 weeks? I forget exactly, but her claim that he took her directly to his house and tied her up falls apart. Or at the very least there is no evidence.

2. She found out about the cameras and the recordings a little late. The existing footage of her entering Desi's house is 100% voluntary, as is any of her initial wanderings around his house. Pretty sure there's also footage of them eating ice cream together.

3. There absolutely had to have been bulletproof footage of Desi entering/leaving HIS OWN house THE NIGHT/DAY of Amy's disappearance. At the very least there had to have been a mad count of 100% bulletproof alibis of him being present wherever else (work, meetings, friends, family, etc etc etc) for a full week or two, during which he should have been busy kidnapping and raping her. Amy also specifically testified that HE (Desi) came over and kidnapped her, not someone "working for him" in case Desi's presence elsewhere could be negated in that manner -- now it can't.

Just trying to make a point that that last piece of the movie was taking the whole "hot girl tricks police" to a completely fictional level. Emotions and a nice ass go a distance, but looking at camera footage and checking for an alibi of a victim in a MURDER CASE is simple routine. It's just something police would do; they wouldn't "take your word for it." And when they did, there would be a case. Maybe she wouldn't go to prison for it by looking sad and innocent to the jury, but there would BE a case.

The first and second halves of the movie were radically different.

The first half of the movie was perfectly logical and plausible. But the second half (after the big reveal that this was simply Amy's plot) was magical and fantastical.

This "first half logical, second half magical" split is very similar to that in Fight Club. Which makes sense, given that both Gone Girl and Fight Club were directed by the same guy: David Fincher.

So to answer your question: In the second magical half of the movie, Amy could simply do pretty much anything she wanted and get away with it. This included murder, artificially inseminating herself with her husband's sperm, and continuing to live happily ever after with a husband whom she tried to frame with murder. I don't think any of this was meant to be logical/believable/plausible.

Note: I haven't read the book and I don't know if the book has a similar thing, or if this was added by Fincher.

I think this is simply a weak point in the plot the movie makers accepted in order to tell the ending as it is. I totally agree with you that police/ FBI or other would investigate about the death of Desi and they surely would find inconsistencies in Amy`s version. Some of them are listed in the answer of Max A.

Personally I dislike inconsistencies but here I think they have their purpose in this story. Besides, the movie already has shown so many details how Amy plans her plausible fake story's. The movie makera could have also added more details on how Amy fakes her claims about Desi but then the movie would last another half hour longer. Therefore, in this case I simply accept the not very plasible killing with no investigation as it is.