Handwave: What problems dealt with and which are just handwaved is seemingly random, although handled pretty well. Nuclear plants and food are mentioned and dealt with, hospitals handwaved partially (the lack of anaesthetic may be OK for dislocated limbs, but people are going to end up with worse) and things like aeroplanes are dodged entirely (for obvious reasons).
Mary Sue: Reese. Though midway through Series 2, she starts developing an arrogant streak that leads to her unintentionally sabotaging the second attempt at bringing the adults back. Huen even more so until he's found out.
The Woobie: Tia very, very much in one episode. And then again a couple of other times. Jeffrey also has his moments.