Re: Pumping excess air into exhaust and petrol catalytic converters
Short answer, not doable.
Long answer, all of the air goes into the engine already. In theory, you could bleed air from the turbocharger. Now, VW exceeded emissions some 40 times. You'd need the turbocharger to supply 40 times the amount of air, use just a little of it for burn and then exhaust all of it. Basically, I've just described a turbofan engine.
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As for catalytic converters -- that's for naturally aspirated engines that use stoichiometric petrol ratio. If the ratio was too lean, you'd wreck the converter quite quickly.
TSI, TFSI, and to a lesser degree, FSI engines use a lean petrol ratio. That markedly improves fuel economy, but at the same time precludes the use of classic three-way converters, and requires the use of catalytic converters for diesel engines. And yes, the implication of there being too many nitrogen oxides also holds true. Don't be surprised to see AdBlue SCR in next generation of petrol engines, too.
BTW, this is what held back the development of lean burn petrol engines up to somewhat recently -- Volkswagen led the charge. We'll just have to see at what cost, and quite soon.