You can choose “customize cards” above your first record and choose which fields you want to see.
The only option you have is that the hidden fields are put behind the “x hidden fields” bar below the visible fields. There is no way to not show them in any way. In other words, they are always accessible.
If we’re talking about the Kanban view, but also the Gallery view, your cards should expand and show you all the selected fields. Are you selecting them in the Kanban/Gallery view itself (and not in, eg, a gridd view)
Not at the back end, no, but the hidden fields won’t show if you share the view.