I fixed it. Android 4.1.2:
Settings > Storage > Default Write Disk > SD storage
This setting was changed from USB Storage to SD Storage.
Now the (real) MicroSD card is seen as **sdcard0 **instead of sdcard1, and Opera can 'see' it. Opera still doesn't recognise sdcard1, but I'm happy with this solution since I don't care to save stuff to the internal memory. I'm still in-the-dark as to why Opera isn't 'normal'.
The fix was far easier than I thought.
Not sure if I changed this setting in my other Android phone since I don't have it with me now. The only downside is some apps that I installed won't work now since they were installed to the internal memory which is now sdcard1 instead of sdcard0, but I'll reinstall them to fix things.
I should probably make a note to change the Default Write Disk to the (real) MicroSD card (whatever it may be called) early on after getting a new phone (before all the installations happen).