

The server owner can choose which channels are displayed on the server page and every user can also choose for themselves if their messages are visible or not. The default setting is that all public channels are visible (which makes sense, since you can already preview discoverable servers before joining, altough this requires an account), the user setting depends on where you are from and if youre account is part of the adult age group. Messages from teens are always hidden and the setting cannot be changed. For adults, it depends on local laws. For example, in the EU visibility is off by default, in the US it’s on.
But, no matter if you’re a teen or an adult, live in the EU or US, you will never see the account that sent the messages on the website. There is no setting for anyone to change that either.


Yes.
For Servers: Server Settings -> Web Presence -> Discoverable Content on the Web has two settings, one for Channel Summaries and one for Channel Messages. Both allow you to select the affected channels or you can flick a toggle for the entire feature
For Users: Settings -> Data & Privacy -> Show my message content from Discoverable channels on web pages
Keep in mind tho that anyone can access all public channels of every discoverable server already anyway, with the change you just no longer need an account to do so, although the web page does only show message content and reactions, not the account that sent the message. To keep your server and its content private you should not have made it discoverable in the first place. And in case you want to change that as well: Server Settings -> Access -> How can people join your server?