Does your primary teaching job pay more than 50k or whatever the minimum is for your nationality ? I'm not sure if it matters, but if it doesn't you are on the special teachers allowance of earning under 50k. The other work won't be on this. Not sure how that affects it though.
You can't get a reentry permit as you don't have a visa.
You'll get another 60 days,although there are reports that people are getting refused a second one. Of course we don't know if they are telling us the full story.
Yep company needs to supply paperwork gor a non-b outside the country and then for a work permit and then for a visa extension in the country. If they don't, walk away.