I think you are misunderstanding channels. They are only meant to differentiate between build types your project offers, regardless of their version. For example one channel can be “Windows” and one can be “Linux”.
Then when you upload your builds with butler, you select which channel a build should go to (“Windows” or “Linux” in this example) and then pick the version number of that build. Then butler will create a history for each channel, and users using the Itch app will be allowed to select a previous version, if they want.
By default the latest version is visible to everyone.
What happens to the channel if I click the “Delete file” button on the “Edit game” page? Does it delete the entire channel, or does the channel still exist but it’s no longer available for download?
I haven’t tried it, but I believe it deletes the whole channel, including all past builds.
What happens to players who are using a channel after “Delete file” is clicked?
If someone has already downloaded that version locally, they’ll still have access to it, but they won’t be able to automatically update through the Itch app.
Is butler able to rename a channel? For example, to rename the 1.15 channel to 1.16? I am aware of the “Change display name” option, which is not what I’m asking about.
I’m not sure, but I would imagine not. Channels are not meant to be named after each version number, that’s why butler has the “–userversion 0.0.0” command, where you can specify a version number for each build, even if they go on the same channel.
Hope that helps.