peakd.com seems to fix my #1 problem (wherein a new tab ignores my logged-in session on another tab). I am yet to know if my problems in #2 and #3 would be solved, although that means I have to move from hive.blog to peakd.com. I know that both are Hive (d)apps and are based on the Hive blockchain, but I learned a lot while I was using Hive.blog.
Thanks for your suggestion!