Well, I removed all files that contain 'Opera' in the filename using 'search,' downloaded and installed a fresh copy of 10.53, and I've done that TWICE, running CCCleaner betweeen installs.
Surprise surprise. Still no Search-Box or Submit button in Opera, still can't install a cookie. Just cycles to Log-out and back.
I suspect something in Opera Speed Dial. It proved tough to remove, and I'm not sure I got all of it. Possibly something that doesn't have Opera in the filname.

I'm giving up on Opera and using Firefox for now. Thanks for the suggestions, all.   Chris.