If one transistor in a circuit (say common source with active load) operate at the strong inversion region, should the rest of the transistor also operate on that same region? Or will it depend?
No, you can't assume all the FETs in a circuit operate in the same mode.
Each FET operates in the mode determined by the potentials at its terminals.
In a simple analog amplifier it's likely that the designer arranged that many of the FETs will normally be operating in the saturation region. But in a CMOS digital gate it's likely that the designer would arrange for some devices to operate in the triode or saturation region while others are in cut-off.