How to add custom user fields to their comments?
I haven't tried this out for sure, so this is kinda just spit balling, but the way I would do this would be to write a small little custom module. So, example:

You could have the date/avatar field stored as a field in the user table. Then in the comments, add both a date/avatar flied to the comment (under /admin/structure/types/manage/[node type]/comment/fields). So now you would have to two places to save the info. 

To hook them up, I would write a custom module that uses hook_comment_presave to intercept the date and avatar field before the comment is saved, load the user info from the commenting user, and then replace the comment date/avatar from the user date/avatar. That way, the information that was with the user at the point of commenting will be forever with the comment. 

You will of course need to theme it, and you will need to hid those comment fields from the user in the comment form. Your also gonna want to make sure the avatar is persistent, but I think this should be a workable solution. At least, that's where I would start.