Once the safenetwork is up and running for everyone does one need to have safecoins in order to visit a safe website or other type of applications build on the network?
No, browsing and downloading files do not require an account or any safecoin.
Only if one posts messages (say email, forum, or whatever) will they need safecoin and an account.
For APPs it depends on the app and if it is storing data of any sort on your behalf. If it does not then no account or safecoin needed. If it does then you do need account & safecoin.
It should be possible though for an app to store data using the app creator’s safecoin. In this special case then data can be stored without your safecoin, but the data is owned by the app (most likely)