Wow, thanks for your detailed response, Milo.
The things Malwarebytes found were called msimg32.dll (Rootkit.0Access), ~!#3027.tmp (Trojan.LameShield), ~!#4186.tmp (Rootkit.0Access), 6ddfe56f-107b773c (Spyware.Zbot) and rtpsa.dll (Trojan.RedirRd112.Gen) (all were supposedly quarantined and deleted apart from the last, for which the log says 'Delete on reboot'.
I suspect the kids may have visited some sites that made us vulnerable when their friends showed them how to download custom 'skins' for minecraft, as the problem's surfaced not long after they did that. We're usually strict on not clicking on things you don't trust, but let them do it this time, to our detriment it seems.
I'll work through your instructions now and see how it goes, and again, many thanks.