No, bird sounds do not influence the release of oxygen from plants. Oxygen exits plants through openings in the leaves called stomata (singular = stoma). While stomata do open and close due to environmental stimuli, the chirping of birds one of those triggers. Rather, signals such as sunlight, CO₂ levels, humidity, and temperature cause cells around each stoma opening to expand or contract, thereby opening or closing the stoma. By following these signals, plants take in sufficient CO₂ when conditions are good for photosynthesis, but limit the loss of precious water. Because there is no sunlight at night, the stomata are typically closed during this time. They then open in the morning when there is again sunlight. That this coincides with the time that birds are especially vocal is simple chance.

Also note that while the stomata of most plants open during the day, plants in especially dry and hot environments have stomata which open at night to prevent losing dangerous amounts of water.

Plants release oxygen during photosynthesis, the process in which plants make food using energy from the sun. I do not think there is any evidence that the process is affected by bird sounds, although bird chirping likely coincides with the sun rising, stimulating the reactions for photosynthesis.

The question of whether plants respond to sound has been around for a while. It used to be trendy to play classical music to houseplants- although there is no evidence that it has an effect! However, scientists have conducted some research that suggests plants might be able to respond to some sounds, such as the sound of running water. It sounds silly, because plants don’t have ears or a brain… but sound is just vibration of the air, and plants can certainly respond to other physical stimuli.

So far as we are aware, no. Trees release oxygen as a byproduct of photosynthesis, not because the birds trigger them to let it go.

What I found is that male birds are defending their territory in the early mornings before there are lots of other sounds around. Supposedly the males do most of the chirping. The scientists thought sound carried farther in the early mornings when the air was more still, but that didn't seem to be true. Their experiment was to play recordings of a white-throated sparrow and a song sparrow at both dawn and midday. They measured the strength and quality at various distances in open grassland and in the forest.

It's just a newspaper article, but here's the link.

In all likelihood false. It's not impossible, but there is little research on this subject. Some evidence suggests that seeds respond to vibrations. Certain plants release oils due to the buzzing of insects.

Thanks for the question, and this one required some digging. TL;DR version: it’s likely a “yes” to some degree.

Now, as that’s likely not as satisfying, I’ll give a quick synopsis on a pertinent scientific discipline: “plant bioacoustics” (if interested, I would search this as there are fascinating studies in the field). To be brief, researchers have played sounds to plants that contained a wide variety of frequencies from a speaker in a controlled environment, and these frequencies are likely found in birds and other animals. The plants exposed to the sounds over those that weren’t experienced changes on the molecular level. These changes were related to events associated with control of oxygen release in plants, such as stomata (pores in leaves that regulate the flow of gases like oxygen and carbon dioxide) staying open or closed, and the rate of photosynthesis. Therefore, it seems that certain sounds played to plants causes fundamental molecular changes related to photosynthesis and gas exchange.