Lewis base strength and the hydrogen proton
One of the keys in the definition of Lewis acids and Lewis bases is that they react to form a Lewis adduct.  To compare the relative strengths of different Lewis bases, you would allow them to react with the same Lewis acid, form the required adduct and measure the enthalpy of formation of the adduct. The adduct with the largest enthalpy of formation would contain the strongest Lewis base.  

1) So Lewis acid and base strength is about thermodynamics.

2) You could run the reaction in any solvent (or gas phase) as long as all of the Lewis acids or bases you want to compare are able to form an adduct in that solvent.  There is no reason why the order of a series of enthalpies of bond formation has to remain constant when you move from solvent to solvent, so your relative strengths could well change from one solvent to another.  The same applies to oxide \ sulphide relative Lewis base strengths if you substitute some other Lewis acid in place of the proton.  Again the relative ordering might well change.