Is it safe removing block-inner clearfix classes from block templates? Is it safe to remove block-inner clearfix templates from block? I use mainly context module which works with blocks. Looking at html there is too much wrappers involved. Thank you.
You can remove them, but they are there so that you can properly float elements within your content. That will depend on the modules you use (whether they include clearfix), and your own CSS.

Not sure about whether Admin requires it. In my case, I use an admin theme for all editing, precisely to avoid such complications.