Assalamualaikaum , Is it necessary to use "Abdul " with the names of Allah. For example, Abdur Rahman, Abdul Basit .. Or we can just name and call a person as Rahman or Basit as well ?

Abdul is only necessary with a few special names of Allah, with the rest of the names of Allah you don't have to use Abdul provided that you remove the "al-" from the name, thus you can use Salam (peace) but not al-Salam (The Peace).

Please see these pages that contain more detailed answers to your question:

Can I keep my new born son name MUHAMMAD RAAHIM?

Meaning of Al Quddus and As Salam and difference between them

Permissibility of Using Gabriel/Jibreel for Muslim Baby Name