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Your three questions are tightly related:
You should not augment the data before splitting. This leads to data leakage, as there is an overlap between the training and the test data, because you are testing your model on some images that have been already seen (although in a transformed version) during training. Therefore, you should first split, then ...
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You only need such a projection if you are using only dense layers for your model (i.e. a multilayer perception (MLP)). You can simply have a convolutional autoencoder, where the layers are convolutions and max pooling, and therefore the number of parameters is drastically reduced with respect to an MLP.
You can check Keras' tutorial on autoencoders, ...
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It is possible. Creating higher resolution imaging with deep learning has been done in several fields. Medical imaging is one of the most common fields.
The general approach is called "learning-based upsampling". It is also called "resolution enhancement" or "super-resolution (SR) reconstruction" in the literature. The goal is ...
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