Norway, a country which has to pay their share of costs of running the Single Market and has to adopt most EU legislation. But not being an EU member, has no say in how the Single Market is run and no input into the EU legislative process.
That is incorrect. Norway has input into how the single market is run and exerts its influence when EU legislation affecting EFTA countries in the EEA is proposed.
Norway is also able to veto any EU legislation.
From the Norwegian Ministry of Foreign Affairs:
According to the principle of unanimity applied in the EEA Joint Committee, all the EFTA states must agree in order for new EU legislation to be integrated into the EEA Agreement and for it to apply to cooperation between the EFTA states and the EU. If one EFTA state opposes integration, this also affects the other EFTA states in that the rules will not apply to them either, neither in the individual states nor between the EFTA states themselves nor in their relations with the EU. This possibility that each EFTA state has to object to new rules that lie within the scope of the EEA Agreement becoming applicable to the EFTA pillar is often referred to as these parties right of veto.
Previously, Norway did not exercise this right, as the governments of Norway believe Norway would be severely weakened if it did. I guess this is similar to the UK Government's position between 1973 and 2011, when the UK did not veto any EU legislation (it did secure opt outs though) in fear of weakening its standing within the EC/EU. Indeed many in Norway complain that their governments have been too passive when EU legislation has been implemented and a minority even want Norway to exit the EEA.
However, for the first time in the history of the agreement, the Norwegian Storting (Parliament) has voted against implementing postal service reforms in 2011, which amounts to a Norwegian veto of EU single market legislation.
Even before this veto, Norway has managed to assert its sovereignty, by passing legislation which violated a certain EU directive.
Sadly, much of the reporting of Norway's relationship with the EU by the British media is of poor quality, which leads to many in Britain to think that Norway has no say in legislation and no veto.
It is true, however, that the EFTA states in the EEA has lost its influence over the 20 years of the agreement, as it has become smaller. The increasing power of the EU parliament and council at the expense of the EU commission, has also caused the EFTA to lose its influence. I understand Iceland want representation in the EU parliament and EU council but the dominant power in the EFTA, Norway, is not interested.
It is likely if Scotland does leave the EU, it would have the same relationship with the EU as Norway does. If the UK (including Scotland) leaves the EU, it would be interesting to see whether the UK, as a great power, would be able to negotiate a better settlement with the EU.