1. Yes, you would file on their behalf (as their representative).
2. The annual gift tax exclusion is per person, per donee. So an individual can gift up to $15,000 (during the 2019 tax year) to a specific donee. Thus, two parents (grandparents or whatever) could give up to $30,000 to each of their children (i.e., $15,000 from each parent) without having to file a gift tax return or otherwise pay gift tax on that amount. Note that the gifts should be separate, not commingled.
