Plants need oxygen when they're not photosynthesizing (e.g. at night).

There are other life-forms capable of photosynthesis and producing oxygen, including many forms of algae and some bacteria. However, if all photosynthetic life were to become extinct, then animals (including humans) would die out as well, yes.

You are absolutely correct, plants definitely need oxygen just like most other organisms. Plants generally have a net uptake of CO₂ compared to O₂. The plant uses this carbon to create sugars (for energy) and construct plant tissue (for structure, this is how plants grow). Oxygen is used to break apart the sugars and release energy when needed. If all plants were gone (let's say that this means all photosynthetic organisms including bacteria), then yes, we would all suffocate from lack of oxygen as it would get used up by all of the organisms left on earth. Nothing would survive, but I don't foresee this happening as there are many, many different organisms that photosynthesize and release oxygen into the atmosphere. Here is the history of Oxygen build-up in our atmosphere: