I'm pretty sure Cable boxes all have HDMI these days, so that shouldn't be an issue.
Monitors won't have an over-the-air tuner, so, you'd need to use an external device to watch broadcast TV.
Monitors won't have an over-the-air tuner, so, you'd need to use an external device to watch broadcast TV.
HMDI from a computer to a monitor or a projector would work fine.