Is resetting passwords that bad, or why was this option even added there? Will the message disappear after 120 days?
There is nothing bad in resetting your password, and it can be good for security if done correctly.Only way changing a password could be suspicious is for account that woke up after long inactivity, drastic changed of writing style, that could suggest account changed hands.
4. I always think of BTT as a highly valuable platform. I'm in fear that my BTT account would get compromised someday if I saved my password in an online place. So I never saved my password anywhere. (I know I should have used a diary to note down my password.)
Diary?! 
Why would you use diary when there are perfectly good open source password managers like KeePass?
Trusting your brain memory to remember all your passwords, or using one-for-all password is worst thing anyone can do.
Note that good password should ideally be long and randomly generated, that is not something you can remember or write easily.