Lapistia

Fired once and then engaged in melee combat.

With the ability to use a weapon in each hand, you can do this in one turn, yes.

Regarding weapon switching, does it cost a turn?

It doesn't cost a turn with Combat Training (there's an extra slot for this).

What if you switch one of the weapons in the pair?

Same thing.

Added after 38 minutes
Fixed the Pain Cult (xD).