These would not be the same in the United States. Could you do my nails? -- This could mean many things: a manicure, applying acrylic nails or painting the nails, for example. Could you cut my nails? -- This is much more specific and limited, meaning only to trim or cut the nails.

Don't be discouraged. Most of these are not language questions but rather references to recent US news stories. Normally, "bath salts" would mean something that you put in bath water for a spa-like effect, but that is not what is meant here. In this context, "bath salts" is the street (informal) name for synthetic cathinones (loosely: amphetamine-like stimulants). These drugs have that name because people disguised them as the legitimate product. I would never use the phrase "eat face" because it is ambiguous, but here it refers to the "Miami Cannibal" attack in 2012. Yes, the attacker here did damage the victim's face and was allegedly under the influence of the drugs mentioned above. I am not sure about "Florida men." It is clearly meant to be pejorative. I am guessing that in the same way that "California girls" attributes specific positive qualities to women from California, this term attributes negative qualities to men from Florida. "Frozen iguanas" refers to unusually cold weather in January, 2018, when the temperature in Florida fell to below 5 C. The green iguanas, which are an invasive species, are not adapted to these temperatures and couldn't move in the cold. Many fell from their perches in trees and were photographed on the ground. "Stand your ground" refers to a controversial law related to self-defense using lethal force (usually a gun). In British common law, which the US follows by tradition, one is usually required to run away from an attacker if it can be done safely. "Stand your ground" laws, like Florida has, dispenses with this requirement and permits a person to defend himself without first trying to flee. The reason for this reference here is that Florida is where George Zimmerman used the law in his defense for the killing of Trayvon Martin in 2013.

To me, a native speaker in the USA, they mean the same thing. I think that "to gain someone's trust" is more common here because it appears more often using a search engine than "to gain someone's confidence."

It means the first lesson is 1.5 times the cost of subsequent lessons. Please note that the expression is "one and one half times" as you wrote in the text rather than as you wrote in the title. Good luck with your studies!