Do item/char abilities which eg. allow you to re-roll a die, also count during the Mythos phase? And if so, does this phase count as a round?
From page 24 of the RR.
A game of Mansions of Madness is played over a series of rounds. Each round consists of two phases, the investigator phase and the mythos phase.
So a round consists of the Investigator Phase, followed by a Mythos Phase. A new round begins at the start of the next Investigator Phase.
This means that an ability to re-roll a die can be used during the Mythos Phase (as long as it hadn't already been used previously during the Investigator Phase).