Re-Scale Training Set
The normalization is recommended as it was empirically observed that
the network converges faster and to better global optima when all
the pixel values lie between (0 and 1) rather than 0 and 255.0. You
can try to see how the network behaves when you train with the
values between 0 and 1.0 and 0 and 255.0.

The mask value does not need to be converted, leave it as it is(presuming you are doing image segmentation), just ensure the labels are correct.

Yes, the same preprocessing that is applied during the training must be applied during the testing scenario/future predictions, otherwise your model will not work.
I'm training a DL model in Tensorflow 2.1. I prepared a training set of several greyscale (1-channel) images.
I have seen that it is recommended to normalize the training set by dividing it by 255.0, so that the values are inside the range [0-1]. So I have some questions regarding it:

Why is this normalization recomended?
My target masks contain values from 0-1 and are float 64. Is it valid to use this data type?
Finally, Should I normalize the images that I use for future predictions as well, or can I just leave them as 8-bit images (0-255).