I know that she was born Princess Olga of Greece, despite being a morganaut. But did she have at least the qualification of Highness? Or none whatsoever?
My opinion is that she was born HH Princess Olga of Greece and Denmark; an unrevoked article of the 1660s Danish Lex Regia indicates that a Danish prince granted permission to settle abroad and his descendants do not need the permission of the Danish sovereign to marry. Therefore, the Danish princely title and the style of Highness descend in the legitimate male line in perpetuity (for now) among the descendants of George I of Greece since he was given permission to settle abroad.
It is not and was not a matter for Constantine II to decide whether or not Princess Olga and her sister are princesses of Denmark and Highnesses or not.
I believe she is a Royal Highness now, however -- ever since her marriage in 2008 to the Duke of Apulia. Correct?
Yes, she is a Royal Highness now.
In this, her situation would be comparable to that of HH Princess Alexandra of Fife (who became a duchess in her own right). By all accounts, her uncle (King George V of Great Britain) was extremely unhappy that she and her sister were granted princess titles in the UK, with the qualification of Highness. But there was nothing he could do about it, since they had been conferred upon the girls by their grandfather, King Edward VII, after he ascended the throne. As sovereign, he was the fount of all honors, and the son had to respect his father's actions.
Didn’t stop him from demoting Alastair of Connaught...
That being said, I'm assuming that George V later on, as king, did approve as dynastic the marriage in 1914 of his niece to his cousin, Prince Arthur of Connaught -- correct? Also, he could not object to Alexandra henceforward being styled as a Royal Highness (as this presumably was in accordance with English common law).
Why wouldn’t it have been dynastic anyway? They were both British dynasts from birth.