I live in Europe and I can tell you without a doubt that the Commission is most certainly not directly democratically elected And the president was not elected by (EU citizen) voters either.
As you can guess from my sig, I do as well. And I can tell you without doubt that there where three major candidates (and some additional ones for other parties) for the commission presidency known before the last EP elections. Juncker (EPP), Schulz (S&D) and Verhofstadt (ALDE). The EPP got most votes in the EP election and hence, Juncker is EC president.
EU citizens did not vote for the Commission or the president or the seven vice presidents. Anything else is not democracy.
They voted for a party who had fielded a candidate for the EC president. Please note that the EC is an executive body, it does not have lawmaking power, that right is reserved for the European Parliament and the Council of Ministers.
If you truly live in Europe you would have noticed our tradition of parliamentary democracy, very few, if any, EU member states elect the executive.
The Commission differs from the other institutions in that it alone has legislative initiative in the EU.
Only the Commission can make formal proposals for legislation: they cannot originate in the legislative branches. Under the Treaty of Lisbon, no legislative act is allowed in the field of the Common Foreign and Security Policy. In the other fields the Council and Parliament are able to request legislation; in most cases the Commission initiates the basis of these proposals. This monopoly is designed to ensure coordinated and coherent drafting of EU law. This monopoly has been challenged by some who claim the Parliament should also have the right, with most national parliaments holding the right in some respects. However, the Council and Parliament may request the Commission to draft legislation, though the Commission does have the power to refuse to do so as it did in 2008 over transnational collective conventions. Under the Lisbon Treaty, EU citizens are also able to request the Commission to legislate in an area via a petition carrying one million signatures, but this is not binding.
You are touching on something important here, the EP and the Council of Ministers does have the power to demand that the EC make a legislative proposal, in case the EC ignores this without a valid justification (e.g. not in the EUs competencies), they can be dragged before the Court of Justice of the European Union. While it would be good if the EP had the power to write and approve motions directly, the current set up is not that bad. In fact, for the constitutional treaty it was proposed that the EP would get this right, but the demands where dropped as the same right would have been conferred on the Council of Ministers as well, the question is whether we would really have wanted that to happen, perhaps it would have been for the better. This said, the EU has been moving in the direction of giving the EP the right of initiative, so it will happen at some point. Small steps...
Regarding the petitions, indeed, if a petition would be in an area which is a national competency, why should the Commission act on it. There is a thing called subsidiarity. First, the petition should be checked for legality, and secondly whether there is any way it will pass. For example, if the EC are told by member states (the Council) that they will not pass any legislation of the petition, there is no reason to continue.
So the public can make a suggestion and the Commission can ignore them and the Council of Europe can make a suggestion and the Commission are free to ignore them. And if you read quotes of Jean-Claude Juncker you can see he has complete disdain for democracy, when he said that if France votes to reject the EU Constitution then they will just carry on regardless and that is exactly what they did. So the EU is not even legitimate, it does not have the backing of the people of Europe.
You can read his quotes in several ways. I.e. when the French and the Dutch rejected the constitutional treaty, you had the following options:
1. Ignore the plebiscite outcomes (but that would be up to the French and Dutch parliaments)
2. Try to find another solution to the issues that the constitutional treaty tried to solve (i.e. too many memberstates resulting in inefficiency due to vetos, ensure that the elected EP had more to say about).
3. Keep the status quo and do nothing.
4. De-evolve the Union
None of the voters where asked which direction to go, they where asked about the constitutional treaty.
The quotes are also mostly without context, i.e. completely worthless for anyone to make a point with.
A system is only democratic is you vote directly for the lawmakers, this doesn't happen with the EU.
The lawmakers are the "two chambers" the European Parliament and the Council of Ministers, the Commission does not have lawmaking powers. The EP is directly elected, the Council of Minister is indirectly elected through national parliamentary elections.
The EC only have the power to propose law, not make it. Further, the EP has the right to demand that the Commission propose a law according to the parliament's wishes.
The US president does not pick the men and women of the Congress or the Senate and he was voted for by American citizens.
This would be like the biggest party in Congress picking a leader for NAFTA, and then that guy dishes out work to a bunch of people who were not elected to their positions, and then those people run all the countries in NAFTA, overruling the White house from Toronto.
If your idea is that the Commission should be picked from the MEPs, why not, it would solve your problems, but this would imply that the EU becomes a federal union. I see no problems with this, but most people seem to be against federation. Note that more democracy (voter influence) at the Union level is the same as federalism for all intents and purposes.
We have two paths:
1. Federalism (more democracy / voter influence, less member state influence)
2. Confederalism (less democracy / voter influence, more member state influence)
Pick one, you can't have both.