is your new company and the old company in a direct relationship? Like main con-sub con that kind of thing. That vendor take up on your current company's job?

Anyway... I think that 30 day is just pure visa. You are not supposed to be working during that period of time right? 30 days is most probably given for people to clear up everything and go back (incase they wanted to leave SG) or for you to look for next job after this one. I'm uncertain about this but you better check.

To clear your doubts the easiest way, why not contact MOM? I'm sure there's some sort of helpline or email address for enquiries.
7.4% remain unemployed... It's actually quite low eh... IMO.

Considering that not many degree graduate actually has over 90% employment rate. But of course it depends on the measure of time. How long is used to define being "unemployed"? Day 1 after graduation? or 180 days after graduation?

Near year end might be a little harder. I'm not sure