"Players start with no extra production of resources."
I'm reading this as "every resource production is set to 0", is this correct? Do players which choose the beginner corp. also set all of their production to 0?
Both correct, yes. (But if the corporation card gives you additional production of something, like Thorgate's +1 energy production, that obviously still applies.)
The only difference between beginner and regular corps is that beginner gets to keep all 10 starting cards for free (and doesn't have any "special ability" beyond that).
Bonus question: If a card has a plant or power tag, can you pay the cost with plants/power 1:1?
No on that one. Only steel and titanium can be used to pay for building or space cards, respectively. (That's why only those have a M€ value printed on their resource boxes on the player boards.)