This sounds most like a camera hardware issue rather than a setting problem, especially since it happens across multiple exposure modes, with different lenses, and with or without flash. If half-press autofocus/metering works but a full press sometimes does nothing, the shutter mechanism or shutter-release system may be wearing out, or there could be dust/grit causing intermittent sticking.

The practical next step is to have the camera checked by a Nikon service/repair center. They should be able to confirm whether the shutter needs replacement or if it’s a minor mechanical issue that can be cleaned or adjusted.