I agree with Philip. In the first, it's because children is in the plural, hence smiles. In the second, it really depends on what you want to say, however, I assume you mean one bus and one train in which case no 2 is correct.
At that time, I was quite happy to see the children smile. ~ The children's smiles made me happy. He comes to school by using public transportation. ~ He takes a bus or a train to come to school. / He takes both a bus and a train to come to school. (depending on the actual case)
Note that it is more idiomatic to say "by public transportation" or "using