Mosfet datasheet reading and power calculation problem
There are quite a few questions.


This is correct. With small pads the thermal resistance is about 300 C/W, so your calculations are correct
The effective thermal resistance of the transistor depends on mounting options, on the area of copper on PCB. The spec lists two typical options, one with ~1 sq.inch of copper area has 100 C per watt. So the max wattage of this transistor of 1.25 W should be considered with best option of 100 C/W. Which means 125 C junction temperature, which is normal.
Apparently you selected a weak transistor for the task. Keep in mind that there is about 200 pF of input capacitance, so there might be transient states where the instant max of 1.25W might be exceeded, and the transitor will blow up.
The graph means that at 100 C the Rdson will be 20% higher than nominal, which would lead to more dissipation and more heating, possibly a runaway condition.
If you drive the gate at 1.2V, the maximum current will be under 200 mA or something.
Yes.


Any more questions?