Yes, specific conditions. For example, here in Ontario, Canada you can employ children as young as 14 but their hours cannot conflict with school hours (including no overnight schedules). And certain jobs are simply 18+, such as working in factories. Abattoirs are one of the more dangerous and stressful (physically and mentally) jobs out there, and would likely fall in that category of hazardous work unsuited to children.

Yes, it's the specific (hazardous) conditions involved. Teenagers can and are legally employed, but there's laws that control the allowed nature and scheduling of that employment. Young teens cleaning dangerous meat-processing equipment during nightshifts is several kinds of nope.

Still happens. It's easier now than it was when I was younger. At 14 I had to get a work permit. Now I would not, the employer just has to report stats to the state.

Our restrictions are also graduated. There's even a sub-14 category, though it's pretty limited. Then there is 14-15, which has the usual restrictions (no hazardous jobs, limited hours per day, no conflicts with school schedule). And then 16-17, which has no limits on hours per day or school scheduling conflicts.

14 is generally the minimum. But there are restrictions, can't work night shifts during school, can't work in hazardous conditions, etc.

Only agriculture or specific types of jobs.