If you quit a web browser, you will see the same reaction.
There is a free password manager called LastPass with auto-login option, this may work for you. I use it but have not tested it within FMP.
In any normal browser, cookies would keep me logged in when I quit the browse, I wouldn't have to log in again unless the cookies were expired or they were deleted.
Yes, but as you have discovered, a web viewer is not a web browser. You can complain about what it doesn't do for you and are welcome to use http://www.filemaker.com/company/contact/feature_request.html to request changes to what it does, but as fellow users, we can only suggest how to get the most out of what we all have to work with.
Yes, I get that now (from my original question I was hoping someone would tell me if this is either expected or unexpected behaviour). I'm just trying to find out what others have done to make it work now.