(Politics In Ancient Greece) Who could hold office?How were the people chosen?

Answer 1

Citizens could hold office.

People were elected to citizenship by being born in their cities.

Answer 2

In ancient greece (classic years) people were elected by voting.
Those that had the right to elect and to be elected were only the ones that were Greek citizens (born by father and mother Greek citizens) and that they had not lost the right to vote as a penalty (see ostrakismos). All Greek citizens had equal rights and it was their political duty to participate in political decisions by turn.
For more info on duties and citizenships read the book :
''L' homme Grec'' (this is the orginal book) I have it in Greek translation but I assume it must have been translated in english as ''The greek man''. It has been written by a group of professors
between which are C.Mosse, O. Murray, J.P. Vernant, M. Veggetti etc