Indeed. The Commission is appointed by the Council (member state govt heads of state) and appointments are ratified by the European Parliament. Same as with the executive in the US. Not complicated.

Euhm. Sort-of. The commission is appointed by country governments (specifically by the minister of foreign affairs). That minister is appointed by a government, which tends to be a coalition government, chosen by the head of the executive, that then has to be ratified by a majority of the national parliament of the country. An appointment to the commission remains valid until their term runs out, no matter what happens to the government that placed them there.

So it's EU commissar (appointed) -> Foreign Minister of member country (appointed) -> Coalition government (appointed) -> Head of the executive of member country (ie. the king, in most cases) -> Parliament of member country (elected)

Certain member countries don't have very stable coalitions, which means there's always a few EU commissars that don't have the support of their host governments. This is especially true now that Christian parties are in the process of being voted into irrelevancy in a lot of countries (unfortunately to be replaced by rightist and in some cases extreme-right parties)

So, what you said. With one very important difference: there isn't one level of indirection, there's at least 4 or 5 levels of indirection. It isn't very democratic, especially since EU law means these people have law-giving power, power to override the parliaments of member countries (that would be why Christine Lagarde proposed using that power to freeze wages)

That's indirect at best. The government also decides who it sends as ambassadors to foreign countries, but would you say that ambassadors are elected?

As far as I know ambassador have a total different role compared to the European Commission.

They sure do, but that's not the point I'm trying to make. We tend to call someone that is elected by the people 'elected'. Someone who gets decided on by these elected representatives we call appointed, they don't inherit the 'elected' attribute. If they did, essentially all political officers (e.g. high level workers in a government department) would be elected, because they were chosen by elected officials.

US supreme court judges get nominated and confirmed. The president is elected (well, kind of, but most agree that he/she is elected by the people even though the electoral college is mixed in there) and then nominates a judge and the Senate confirms the nomination. I don't think it's common to call the supreme court judges 'elected' though, despite their selection by two elected parties.