How could Obito transport his body parts in another dimension without bleeding?
Obito uses a technique called Kamui.


  Kamui is a powerful Mangekyō Sharingan dōjutsu that creates a unique and specialised form of space–time ninjutsu. It allows the user to achieve two very distinct, yet closely associated feats — teleportation and intangibility.


Intangibility, here refers to making the body or certain parts of body inaccessible to the physical world. Thus, no one could touch him and passed right thought Obito.