Yes, with people as objects it does work differently. You can use either "make" or "had" in those sentences above, with the teacher and the dad. Just differs in strength.
For the certificate, these would both be right:
I thought it would be nice to HAVE the certificate checked in order to make sure it is genuine.
I thought it would be nice to MAKE the agency check the certificate in order to make sure it is genuine.
However, "make" suggests a command or force and may sound too strong here. It's OK for teachers, parents, and such but not so good for workplaces. I can "make" my computer program check the certificate, but I would "have" the institution or person check it, or "ask" them to.
I thought it would be nice to have the boss check the certificate in order to make sure it is genuine.
I thought it would be nice to ask the boss to check the certificate in order to make sure it is genuine.
I wouldn't say
I made my car repaired.
But
I had my car repaired. (someone else did it for me)
I repaired my car. (I did it myself)
