1. Ideally, MOSFET have 0 resistance when on i.e. saturation and infinite resistance when off. In practice, it has nigligible on resistance and very high off resistance. Using ohm's law is one way to find out the resistance. but generally linear region resistance is also very small. Look into the datasheet to find all these parameters.
2. MOSFET is a voltage controlled current source i.e. transconductance device, a small variation in input voltage make large change in current through MOSFET. Voltage control is done by gate voltage. More is the gate voltage, more is the induced charges between source and drain, so more is the conductance, so resistance decreases. So, when MOSFET is fully on, the resistance is negligible, the conductance is highest. The threshold is the minimum gate voltage required to make MOSFET start conducting and decrease the resistance. So, drain to source resistance is inversely proportional to gate voltage, its gate voltage function. Since, the resistance becomes very low when threshold is crossed, hence, the amount of conductance increases, so we talk about conductance of drain to source or transconductance from gate.
3. MOSFET's are used mostly as switch, so when mosfet is on, there is almost 0 resistance across it and high current flows, since the resistance is almost 0, the amount of voltage drops is less too. Again in OFF state, there is no current flows across mosfet, but full voltage appears across drain to source. So in both on and off state, the overall power usage is very low, while in linear region, both voltage and current exists due to limited resistance of mosfet hence, power dissipation occurs.
hope that helps.