Yes, it is very common to create a logo for identification of your brand. In US practice the mere "use in commerce" of the logo in association with your goods or services may give you some state and federal protection.
However, a registration (state or federal) will give you better protection, not to mention notice to the public of your claim to proprietary rights in the logo.
So, to answer your question, yes you can "put trademark" on a logo, which means you can enforce the logo as a trademark, or you can put "tm" on the logo, or you can register it and put the circle-R on the logo.
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The circle-R ® registered trademark symbol can only legally be used on a trademark that is federally registered. Many non-registered brands use "tm" or "sm" (for service marks) to indicate a claim to the value of the brand.
Patents have nothing to do with it.
No; it is too short to qualify as a literary work, and as a work of the federal government it is exempt from copyright protection in accordance with 17USC105.
No, for two reasons: as a government work, it is not subject to copyright protection, but also as such a short phrase (one letter!) it could only be protected as a trademark. Which it isn't.