The wudhu is not valid with the shoes on. Praying with the shoes on is permissible if one can place the tips of the two big toes on the ground while wearing the shoes.
If the current authorities of the two holy mosques permit Muslims from all sects to perform their Hajj based on the teaching of their own sects, then one must follow what the religious evidences prove in determining the time to perform the Hajj accordingly. If the authorities have enforced a set day, then it is acceptable to perform the rites of Hajj accordingly.
There is no need for uttering the intention or saying it out aloud. Rather, it is recommended not to talk - even by pronouncing the intention - during the Adhan or Iqamah. The dislike of this is stronger in the Iqamah, and even greater if one speaks after the phrase "Qad Qamatis-Salah" is recited. It is recommended to repeat the whole Iqamah if one talked during it. People's practice in pronouncing the intention originates from the fact that the intention is a condition in worship, without understanding that there is no need to pronounce it.
It is prohibited to react with rapture to the music and the songs; it is not prohibited if the reaction is to the content only.