Pressure alone is sufficient only when the substance is below its critical temperature. The critical temperature for methane is 190.6 K, some 107 degrees below room temperature. Gases that are highly compressed above their critical point form a supercritical fluid, which can have liquid-like density but does not have a defined volume.

LNG is a true liquid, stored at cryogenic temperatures and low pressure.

No. It is not practical to liquify by compression. See: https://www.youtube.com/watch?v=QgtSoEJD9HE

It seems it's hard to liquify by pressurisation.