1. Yes, mathematicians keep notebooks. Sometimes, after their death, notebooks are published. Here is an example: http://www.claymath.org/publications/quillen-notebooks

Here is another example, though not so recent: https://www.math.uu.se/collaboration/beurling/unpublished-manuscripts/

I believe many mathematicians have a lot of unpublished stuff. Sometimes their friends and students publish this after their death, exactly as it happened in the past.

2. Mathematicians discuss on their blogs and web pages not only published results. There are many examples. Much material exists in the form of correspondence, lecture notes, and preprints which are not officially made public (not available to everyone, like blogs) and which mathematicians share among their colleagues.