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Yes you are right, there are a few more reasons offered as to why we wear a Kara

From http://www.bbc.co.uk/religion/religions/sikhism/customs/fiveks.shtml "A symbol of restraint and gentility. A symbol that a Sikh is linked to the Guru. It acts as a reminder that a Sikh should not do anything of which the Guru would not approve. A symbol of God having no beginning or end. A symbol of permanent bonding to the community-being a link in the chain of Khalsa Sikhs (the word for link is 'kari'). The Kara is made of steel, rather than gold or silver, because it is not an ornament."

From https://www.allaboutsikhs.com/introduction/the-sikh-identity-5ks "the kara is also on the right side, which is the hand [with which] most people perform their deeds [ and] is a constant reminder to perform good deeds. By wearing it on the wrist, it binds the wearer to the will of God, and reminds the wearer to never extend one's hand for the performance of evil. The Kara acts as protection if someone goes to strike you with a sword on your wrist. According to scientific research, the Kara adds to the iron levels in the body by rubbing on the skin. The Kara teaches us that these arms belong to Sahib Sri Guru Gobind Singh ji - with which we are not to steal, con, commit forgery, oppress, bully, persecute, sin or murder."

As for non-Sikhs wearing the Kara, I'm not sure. Maybe someone else can offer an answer as i have never seen that.