When the detective is revisiting the home and possible crime scene, the hole in the wall is small, then when she is shining the flashlight in it the bottom portion (where she would hit it with the crowbar in the following scene) is larger. When the camera angle changes, the hole is back to the small size until she hits it with a crowbar.
On the California marriage license Alice looks at while Peter is away, both bride and groom have the same last name. California licenses utilize the maiden name of the bride.
Alice sends money to the detective to give to the victim's widow. Police receiving money would turn it over to the agency and it would be considered found property, with a court order required to turn it over to anyone. There is a good chance the money would end up with the government since it was obtained as a result of a crime. What Alice should have done was mail it to the widow directly.
Alice reads a headline stating Peter was arrested for killing his wife. In the subsequent scene, the detective tells Alice Peter was acquitted (something that only happens at a trial), he had a solid alibi, there was no hard evidence against him, and he had passed a polygraph. Given the alibi and the lack of evidence alone, he would never have been arrested much less gone to trial.
A detective tells Peter his fingerprint was found on the bullet that was removed from Alice's skull. When a bullet travels down a gun barrel, it gets very hot and also has rifling grooves imprinted on it. Fingerprints are left by oil on the skin. The oil would have burned away due to heat, and even if it had not, any fingerprint would be illegible due to the rifling marks. This item was also overlooked 3 months after the crime when the detective talked to Alice at the park- no mention was made of a fingerprint on the bullet.
The bug on the inside of the lampshade was more than big enough that it would have shown up as a dark spot on the lampshade the moment the lamp was turned on. But the lamp was on the entire time, and the bug was never visible.
The bug planted on the inside of the lampshade only started beeping when it was discovered. The volume of the beeps were loud enough that it would have easily been heard if it had been doing it the entire time.
People under arrest at a hospital are guarded 24/7 by officers, they are not left handcuffed to a bed. As such, Alice's escape would not have been possible.
Peter refers to his dead wife as his ex-wife. Since she died while they were married, she was not his ex-wife. The line should have said his late wife.
The police know the gun found was registered to Peter, indicating it was in some sort of database accessible to the police. The fact Peter owned a handgun was never brought up during the three months the detectives investigated the case. His owning a handgun would have been a significant factor in the case.