Oxygen itself isn't flammable, meaning Beth's makeshift flamethrower wouldn't work.
When Jacob rescues Alice/Beth, he injects her with an EpiPen in her shoulder, which isn't the correct way; it has to be injected in the upper thigh; if not, it can lead to serious side effects.
The license plate Mary and Luke find has "Gotham" as its only geographical identifier. Gotham is established to be in the United States. In the US, license plates are identified by state; some states' plates also have a county name but all have a state name.
When Kate gives Cartwright a sip of water, there's no reason for her to set the glass down where she does other than to set up what follows, when he breaks free from his restraints and uses it to attempt suicide.