You can use a mercury thermometer to accurately infer whether mercury expands linearly with temperature
No! You cannot use the thermometer to verify the linear expansion of mercury. This is because the temperature scale itself is defined by making equal markings between 00 C and 1000 C, assuming that mercury expands linearly.
In fact, scientists have now established using statistical physics and other concepts that mercury does not expand exactly in a linear fashion.
Before all this, the only way to guess if a thermometer fluid is a good choice (i.e, expands linearly) was to observe which of them gave consistent results.