Is oxygen always necessary for the production of ATP in your cells?

Yes. Part of cellular respiration (ATP making) is the ETC (electron transport chain). As the hydrogen ions move throughout the ETC, it eventually comes to a close. Just as if a whole bunch of cars were driving on the road, and the light turned red, all the cars would have to stop. The same is for the electrons. Thus, oxygen is REQUIRED to keep the ETC flowing. If the ETC can flow, then the organism can go. That hydrogen ion (electron) bonds with the oxygen to make...water!

That water than powers the ETC in plants for photosynthesis, by the splitting of water. Then that released oxygen is used in CR!
It may be more than you need, but hopefully I answered your question! Good luck!