The 'Multi class rating' (officially known as the Multi-Engine Piston Landplanes or MEP class rating) is the basic qualification to fly multi-engine aircraft. It is valid for all multi-engine piston types weighing less that 5,700kg, with a few exceptions (certain bigger/more complex types, and seaplanes).
The 'Multi-engine instrument rating' (Instrument Rating Single Pilot Aeroplanes Multi-Engine or IR-SPA-ME in JAR-ese , often known as the 'ME-IR' or 'multi-IR'), allows you to fly multi-engine aircraft in meteorological conditions worse than those permissible for flight according to Visual Flight Rules, and in airspace that requires flight according to Instrument Flight Rules. As its name suggests it is valid for all single pilot multi-engine types, it is also valid in all single pilot single-engine types.
As for the 'Multi-instrument rating completion' - maybe the school breaks its ME-IR course into two modules (maybe sim and aircraft), and is quoting both seperately here?
As has been said, the instructor rating isn't necessary for an airline job, but is necessary if you are going to follow the instructor route into the industry.
cheers!
foggy