Should I trust my employer with my passport if they need it for certain processes?
There will probably come a time when your employer needs to show your original passport as part of the visa / work permit process, etc and photocopies won't suffice.
If at all possible, ask if you can accompany the school staff on these processing trips. In other words, don't let your passport out of your sight.
There have certainly been instances of school admin staff losing passports and while it isn't the end of the world, it can be a real hassle and inconvenience to get a passport replaced.
Asking to accompany the school staff on a processing trip is certainly not an unreasonable request.