Video classification of birds
Are there better techniques for doing video classification compared to using temporal segment networks?

I would suggest to train the network to detect bird species in image space without the temporal dimension. This will make the network have much less parameters to tune and you will likely get better results. Then for a novel video, you can detect the species for each frame and do a majority vote to decide what species was in the video.

Are there any preprocessing steps that should be done to improve accuracy?

If you train in image space then you can use a DataGenerator on the image and transform it in all kinds of ways, the position and rotation of the bird does not affect the species. This can be done through the Keras pipeline. You can check how to do this in general here.

How to train the model to generalize better to be less sensitive to lighting and weather?

You can try to pass your images through some pretrained segmentation network to identify background and foreground and then set that area to a static color. Perhaps black (i.e. all 0's). This will make the decision boundary for the network easier. However, I think you should focus on the other points, the network should inherently detect what features are the most salient for discriminating different species, if you have sufficient training samples the weather and background will be low information features and will thus not be used.

How can the model be trained to classify unknown species?

This would be called anomaly detection. This is a much harder problem. If you have birds of other species available in your dataset then you can add a class label for other. This is still standard supervised learning. If this is not available it will be very difficult. Most anomaly detection algorithms do not work well for high dimensional datasets. Then you will need a way to compress the information in your input images. This can be achieved with some transfer learning on a pretrained autoencoder network trained on natural images such as ImageNET, then you can use transfer learning using your bird images to further tune the network for your specific use case. You can then use this compressed feature space using one of the anomaly detection techniques I described here.