1. The US exit code (011) is definitely not needed when using your US cell phone abroad. An exit code is however needed to dial to another country. For example, all European countries use 00 instead of 011.
Indeed, your phone becomes a local phone for calls made to local numbers so just dial what locals dial.
2. All sources are correct: the plus sign replaces 011 for international calls originating in the USA. Therefore you may use + or 011 (never both). Landline phones do not have a + key so you’d have to use 011.