Can Zebra Finches spread diseases to people?

I've never been a fan of indoor birds because of a fear that they will spread diseases to people. However, I'm trying to get over it, as my kids absolutely adore the birds. We just got three Zebra finches (1 male, 2 females) for free from someone who didn't want them. I'm researching proper care, feeding, lighting, etc.

What I'm really concerned about is whether or not having these birds in my house can put my children at risk for diseases. Are they the type of birds to do that?

Bonus question: I have the cage on top of a very old full size freezer with the black grating behind it. Will the feathers get stuck in the grating so badly that it will break the freezer? They're placed here because my children are preschoolers.

Thanks for the information!