How does moving up the zones in Manhattan work?
You only move to the next zone in Manhattan during your Move Phase, after resolving dice and before buying cards. If you're playing with 5-6 monsters, both monsters move along their individual tracks (i.e. the first monster would move along the 2-4 track, and the second monster would move along the 5-6 track).
When do I collect my reward?
You collect at the beginning of your turn, then you roll dice, and move to the next zone after resolving their effects.
Can I heal while in Manhattan?
You CANNOT use any heart dice to heal yourself, however they may still be used to remove curse tokens or activate any other card effect. You CAN heal by using cards and by destroying units that reward you with hearts.