Yes, diagonal movement is permitted.
Attacks can only take place if the gladiator is in range (i.e. in the same zone unless using a Sagittarius, Velite, or Hoplomachus' special moved) at the start or end of the round.
EXAMPLE: A thraex starts the turn in Zone A, and a murmillo starts in Zone B. If the thraex steps into Zone B to attack, but the murmillo steps out of the zone in the same turn, no attack can take place.