About democratic legitimation:
The members of parliament have the mandate to elect a head of government in most European states. This means that such prime ministers are democratically legitimated in the framework of representative democracy, even without having been candidates for the job during general election campaigns.

About the Greek guy:
Poor choice.
He already proved his lack of integrity and lack of competence to no doubt.

What the Greeks really need is someone who leaves the Euro, converts all debt to free exchange rate Drachme, inflates the debt away and by the way also breaks the sticky wages that inhibit many reforms.
Greece could be out of its troubles by 2020, if only it leaves the Euro and gets rid of the debt. It might become unbearable as EU member by doing so, though.
Maybe it could let its membership rest (= not being subject to EU rules and decisions, but also no Greek bureaucrats in Bruxelles, no Greek EU parliament or commission seats and no Greek vote in the councils).