I've lost track after jaden yuki's episodes and I don't know anything about Zexals and Pendulums but I have one question about the game... But I was playing like hardcore before.

Why the first player guy doesn't draw a card at his very first turn?

is it a new rule that comes with the new systems or is it a bug?

It's a new rule, they did this because there was no reason for a player to start 2nd.

Also, another new rule is that there can be two field spell cards face up simultaneously (one for you, one for opponent). Also, when you active a new field spell card, the old one is SENT to the graveyard and not destroyed (so, for example, Geartown's effect doesn't trigger anymore by just activating a new field spell card)
Thanks a lot for the quick response. Although it was convenient to start second if you are using cyber dragons but ... the logic you mentioned is reasonable also.