Setting Lineups for The International 4 Fantasy Leagues
Solution 1:
I'm in the UK, and the start time for me is 1AM, that is the time that the roster locks. I assume that's the same time worldwide, I recall this being part of the fantasy league FAQ on the dota2.com blog, but I'm unable to check from this PC.
Schedules is supposed to be an indicator that the player is sheduled for non-amatuer tournament games that day. I think there were some bugs in this when it was first released, there were some notes about it in a recent changelog and it seems to be more consistent now, but like everything, will be fixed properly in time i guess.
As T_O mentioned, I would use the JoinDota listings for matches as there doesn't seem to be a proper comprehensive list in game, or the watch tab if you want to trawl through each tournament by hand ;)
There's a schedule. If you're +/- UTC, assume it's not catered to you. Trial and error can improve this answer...there's an edit button if you're here during TI5.