On one hand you say: the holy Prophet did not select any successor, neither did he order anyone to select a successor for him, and people chose AbuBakr to be the successor;AbuBakr also appointed Omer as his own successor, Othman was also determined by the six-member council to succeed Omer. On the other hand you argue: these were successor of the Prophet and refer to them as Caliphate Al-Rasoul. Isn 't it a statement of lie about the Holy Prophet? According to this Hadith-Al-Motevater: (من كذب عليّ متعمّداً فليتبّؤ مقعده من النّار) Sahih Al-Bukhari vol 1, p 36, 2/81, vol 4, p 145, vol 7, p 118. رواه من الصحابة ثمانية وتسعون نفساً said Ibn Al-Joza. Almodou 'at vol 1, p 57. رواه مائتان من الصحابة ، شرح مسلم للنووي said Al-nouvi, Sharh Mosallam Al-Nouvi, vol 1, p 68.
Any slander about the Prophet is a sin. By considering that you are right when you claim that the Prophet did not select a successor, Rashedin Caliphs are not caliphates of the Prophet!