Airfield QNH is measured by taking from a calibrated precission instrument the observed pressure at aerodrome level. This is then adjusted by the Met Observer to sea level pressure by applying a correction. The correction allows for deviation from the ICAO Standard Atmosphere (ISA) by taking into account prevailing temperature and any extremes of pressure.
This corrected value is given to ATC who will pass it to the pilot who should be able to set it without having to make a further correction.
No and no!
The value given to the pilot is
not the sea level pressure, it's the QNH. The sea level pressure is used only by meteorologists in plotting charts at a uniform level.
On a day with any temperature deviation from ISA, a particular QNH will result in a correct reading of altitude at only
one level. The level that is chosen is the aerodrome level. Thus at any level
above the aerodrome, an altimeter reading using the aerodrome QNH would be subject to temperature error, and the pilot must correct critical altitudes accordingly.
(The idea of using an old altimeter to obtain QNH is antiquated, it might still be used by some but IMHO they really should know better?)
It makes no difference in principle if you use an altimeter to measure QNH or whether you use a "calibrated precision instrument" to measure QFE and then use a look up table to derive the QNH. Temperature is not involved provided you measure at the aerodrome level. Of course if you make the measrement at the top of a 100 ft tower and then deduce the pressure at aerodrome level, you'll need to make a small correction for temperature.