Then, when I set this option back again, to how it had been, the problems did not return.
Probably because the necessary data was already downloaded?
Possibly. Anyway, this needs updating.
It does seem that, for whatever reason, the latest version of Opera (Ver. 39.0.2256.48) does need the option: 'Block third-party cookies and site data' to be disabled. Otherwise I get two problems on one website I use - Firstly, very slow loading of pages. Secondly, some site icons do not display correctly, though they still work. (And they display correctly with the 'Open image in new tab' menu option.)
AFAIK this was not the case in previous releases of Opera. Why this should be so I do not know. Perhaps something has changed on the site? Perhaps the option did not really work as intended on previous releases of Opera? Perhaps there is a fault with the latest version of Opera?
But something has changed, there can be little doubt about that. I have previously used Opera with this option set and not encountered problems previously with the sites I use.