Tried both - the problem still exists.
I took more radical steps - I uninstalled Opera from my notebook (and choose "Delete all personal data" - because otherwise even freshly installed Opera does show my bookmarks in the Bookmarks bar - and I did tell it Not to import data from default browser ) prior to installing a new copy to "Opera Test". Freshly installed Opera (No extensions - except installed by default Opera Ad Blocker - which is disabled) worked just fine, I was able to download all attachments from a message in Gmail.
But once I geared Opera up and told it to "Synchronize" the problem popped up again - in synchronized Opera I keep experiencing the problem with inability to download attachments. And I need synchronization to work across two of my notebooks - one stays at home and another goes with me wherever I go.
When synced, I can see my bookmarks populate the bookmarks bar. No extensions, default Opera Ad Blocker still disabled.
Signing off of my account doesn't solve the problem - once it popped up, no matter whether I' synced on not doesn't matter - attachments won't download.
Then I repeated this with Opera Developer version - by uninstalling Opera (and deleting personal data) prior to installing a new copy to "Opera Developer". Everything was the same - on freshly installed Developer version Gmail works just fine, then I Synchronize and voila - the problem is back and couldn't be fixed by signing off.