Okay, for your first question: for the way this particular plot is set up, only the comms and the holodecks will receive the time/space calls. However, given that the tech in Kaylin's world works differently, we're inclined to say that even though the call is coming through on the comms, her father is using his mirror on the other line. If you want to say Kaylin's mirror is lit while the conversation is happening, sort of like how a cellphone stays on even if you're talking on bluetooth, then it can be handwaved that Q just sort of re-routed the mirror for his own means.

Regarding gifts for Q, at the end of the week there will be a post where gifts can be left and Q's presents in return will be sent. What your character leaves for Q is entirely up to you. It can be something meaningful to your character, something your character genuinely thinks he'd like, socks, a tie... let your imagination be your guide, so long as you're prepared for him to grouse if the gift isn't up to his standards. ;) Bear in mind that these gifts can't break the in-game rules!