Does closed-back mean active noise cancellation?
No. Closed-back describes the housing; ANC is a separate feature. For example, the DT 770 PRO X uses passive isolation and has no ANC.
Can one pair handle gaming and podcasting?
Yes, with a suitable playback source and separate microphone when required. If you regularly record, start with closed-back options and check leakage in your actual setup.
Are open-back headphones always better for mixing?
No. A quiet room, familiarity with the headphones and how your mix translates elsewhere matter more than relying on one label. Use familiar reference tracks and check your decisions on other systems.