Because protection is automatic, a notification is not required: everything is protected by copyright unless it specifically says otherwise.
Typically this type of usage would be licensed; there is rarely a fee, but rightsholders do expect to be listed on the copyright page.
That would be copyright infringement if the original song was protected.
Assuming you have a license from the copyright holder to print the song in your book, you would hold the rights for the book as a compilation.
If you are not the composer of the song then you can NEVER take the copyright as yours; you would have to purchase it from the copyright owners. Also, when the copyright expires, 50 or more years after the author's death (or after publication, depending upon circumstances and national laws) it is no longer copyrighted at all and nobody can possibly own the copyright.
If the underlying work is protected by copyright, you would need permission from the copyright holder to arrange it (the law calls this a "derivative work," and publishers call the permission "print rights"). The resulting agreement would stipulate who gets the rights in the resulting work, and it's almost always the copyright holder of the underlying tune. If the underlying work is in the public domain, such as a folk song, then yes, you would control copyright of the arrangement. However the original song would still be in the public domain.
Yes
A copyright.
No, song covers are not copyright free. Covering a song without permission from the original copyright holder can lead to legal issues.
Without a license, yes. That is assuming you did not personally compose, perform and record the song yourself, or take a public domain composition and perform and record it yourself, either of which would mean you own the copyright on those recordings and it would not be copyright infringement to use the recording of the song any way you like.
There are at least three copyright holders, but for most uses I would suggest starting with Universal.
Yup. Translating in another language is almost the same as covering it, because the melody keeps the same. And a song with copyright you can't cover, its against article 21 paragraph 4 of the copyright law of USA.
Of Coarse! If not anyone could steal her song from her and sing it and make a profit and it would be legal. So of Coarse it has a Copyright.