Yes, all of them do nowadays anyway.
Yes they do back ground checking but its just on the previous work you've been
They do background recheck base on what your performance to your past employer, it takes a week to accomplish their checking
Yes they do background checks as soon as they can process papers.
While I was applying in this company, They did not require background check. They rely what the resume's information and did written examination and interview. It did not take a long time to wait but after the exams and interviews, they hired me immediately.