Are they authentic honeywell ?

I am a hospital worker and we use Honeywell DF300 N95s. I've done a very close comparison and, as best as I can tell, these are genuine Honeywell N95 masks. The markings are identical, the mask pattern is identical, the way the bands are stamped onto the mask are identical, the packaging and all the package markings are identical, and the internal (open) plastic bags are identical. They feel the same, and the weight is the same. IF they are fake, then these fakes are as good as the originals. I'm comfortable buying them. Yes, I get them free at work. I bought several boxes for family members.

Is the plastic "foam" over the nose piece closed cell or open cell?

Close cell, great mask, easy to breath, seals all the way around the face, unlike cheap masks.

Is it approved for children

there are no approved masks for children. but my 10yo who is big enough to fit an adult kn95s can wear it daily to school. however, my 5yo uses south korean approved kf94 masks and they fit well. Make sure they are made in South Korea (not China, they make a lot of counterfeits) if you'd consider.