Which reveals that data-free inferred temperatures at 2 m are fairly worthless whereas those at 925 hPa are probably better but not necessarily optimal nor supported by much actual measurement

Compare the current +80N T2M from ECMWF and from the NOAA CFSv2. These are two completely independent systems using different model physics and assimilation methods yet they produce remarkably similar (though not perfect) results. "Worthless" depends on your perspective and intended use of the data. My guess is that it has been substantially warmer than normal in the Arctic. Summer results are a different case.

The map below shows the region over which the mean 925 hPa Tair is computed. The 00z, 06z 12z and 18z [time points at six hour intervals] analysis fields [spreadsheet columns] are used for making the daily average temperature

In NWP an analysis field is one in which the background/forecast model variable (a best guess) and the observations are combined to give an optimal representation of the state of the system i.e. the result of assimilation. For global NWP, an analysis field is typically calculated 4 times a day, with the 00z and 12z usually having a greater number of observations. The analysis fields (e.g. surface pressure) of model state variables form part of the initial conditions of a model forecast. For times outside the analysis, results for the variables subject to analysis are effectively forecast values from the model. Variables not subject to analysis (e.g. solar radiation at the surface or T2M in CFS) are always forecast fields.

925hPa temperature .... should be more influenced [improved] by data assimilation [what data?].

As already mentioned ... Numerous satellite soundings (HIRS, MSU, TOVS etc.) are used to solve an inverse problem and provide information about the temperature and humidity of the atmosphere. Radiosondes provide information from coastal locations. Scatterometers can inform about winds (over open water). Aircraft taking a polar route provide info. Surface pressure measurements give information about the mass of the atmosphere etc. In the CFSv2 all this data is put into a 3-D variational assimilation system, meaning that a measurement at one location (and/or level in the atmosphere) will influence other locations/levels. This influence happens at the time of assimilation (through spatial covariance and model re-balancing) as well as through the spatial and temporal propagation of perturbations in the atmosphere.

The +80N 2 m Air Temperature is a diagniosed [sic] rather than a prognositc variable [sic] [intended meaning: calculated retrospectively from models and meagre observations]

In terms of modeling, a diagnostic variable is one that can be computed from the current set of states and fluxes (e.g. T2M, which is also a special case as it is generally not used by any other part of the model - it is essentially a post-processing output). A prognostic variable is one that (in the case of NWP) is calculated by integrating an equation forward in time - for example, you iteratively solve the surface energy balance to prognose the surface radiating temperature. The dynamics & thermodynamics of the atmosphere are solved (prognosed) on a grid; the 2m level is not one of these grid levels. Once you have prognosed the temp, humidity & winds of the main atmosphere, and solved the surface energy balance, you can then diagnose the T2M. Prognostic and diagnostic variables may or may not be analyzed fields (i.e. model + obs merged).

T2M analysis at ECMWF is an optimal interpolation [an older method for choosing weights to minimize the mean square error, a supposed minimum variance estimator]

ECMWF T2M assimilation is Optimal Interpolation (OI), but the main atmosphere model at ECMWF uses a full 4-D Variational method, meaning that it not only takes into account of the horizontal and vertical state of the atmosphere, but also looks at the trajectory of the model in time (compared to the temporal trajectory of observations). This is currently the most advanced system in the world.

You need to choose the appropriate tool for the job - sometimes a plain old screwdriver is just as (or more) effective as the latest power drill/driver.

Hope that clears up a few things ... now, back to your regularly scheduled programming