The details are unique, but not the story
Any Singapore-registered business can sponsor a foreigner to work here, up to certain quotas. It doesn't have to be home-grown, of course.
Why worry about your first question, when the firm is supposed to apply the EP on your behalf, and you must have it approved before starting work.
On whether there's double-taxation or not, please visit
www.iras.gov.sg where luckily Malaysians don't get it
I think the best resource for your 3rd Q is
www.mom.gov.sg itself.
Goatboy will always cherish his former goatgirl.
But the world is full of fluffier ones.