Where does Starbucks Europe pay taxes?

It's not customary for a businesses to complain to the European Council. They can lobby it for new rules but its role is limited. The regular procedure to dispute a Commission decision of the kind you are alluding to is to bring the matter to the Court of Justice of the European Union (first to the General Court and, potentially, to the Court of Justice, which acts as a kind of appeal court in such cases).

The different steps of the Starbucks procedure are detailed on the Commission's website. The General Court annuled the Commission's decision in 2019 (full text of the decision) and the Commission apparently decided not to appeal that decision (interesting analysis). Starbucks won't have to pay back taxes to the Dutch government and the tax ruling presumably still stands.

Note that the Commission has taken a more activist stance on tax issues in recent years but it is limited in the legal basis it can invoke. In particular, it is not competent to review member states' tax codes or whether a particular Dutch or Irish tax scheme is unfair or beneficial to the EU as a whole. All it can do is check whether Starbuck got a better deal (through what is called a “tax ruling”) than other competing businesses could have gotten from the Dutch tax office, thus distorting competition.

That's why you will see that in these cases (the famous case against Apple follow the same basic framework), the Commission is not taking a decision against a business but against a member state and its tax authorities. The notion is that the tax ruling amounts to an illegal state aid for the business in question but not that the business cheated any of the local tax rules. The Dutch or Irish governments are therefore also a party to these cases, arguing that they should not receive any money. The companies are however allowed to launch an appeal of their own because their interests are potentially harmed by the Commission decision (they are the ones paying the taxes after all) and the court usually join both cases together.

Consequently, when the court upholds a decision (e.g. in the Fiat Chrysler case), the company has to pay taxes to the country where the tax optimization scheme is based (the Netherlands, Ireland, Luxemburg…) and not to the countries where the bulk of their retail operation is located (UK, Germany, France…). Ultimately, this is about tax authorities in different EU countries undermining each other and making these schemes less lucrative but not directly about how much taxes are paid locally (even if that's probably in the back of everyone's mind).

This also means the Commission cannot do much about transfer prices, whether or not Starbucks has to pay corporate taxes in the different member states or the general structure of the scheme they are using (the low tax rates for royalties in the Netherlands). All it can act on are tax rulings guaranteeing a low(er) rate in advance to a specific company.