The cost to airlines of ETS will come from the purchase of credits over and above their free allowances. They can either buy them on the open market (from other airlines, other firms in ETS-governed sectors, or brokers) or from Governments who will have held back 15% of the available aviation allocation when giving out the free credits. The money will not go to the EU.
‘Earmarking’ of revenues for environmental projects would be nice, but not necessary from an economist’s point of view – he would say simply that what is important is that the price of the product reflects its external costs (environmental damage etc) as well as its production cost, thereby allowing the customer to make a rational decision whether or not to purchase. In this Utopia, revenues could just as easily be spent on healthcare or education.