Backup restore never completes

Several months ago I noticed something amiss with the interaction between my iPhone & iCloud - first inability to complete backups to iCloud, then lost iCloud-based reminders, finally a persistent sluggishness of anything related to iCloud settings, most visible when accessing iCloud settings from Apple ID screen (would take about a minute) & when either showing the list of available calendars or selecting a calendar for an event being created (either action resulting in frozen screen for minutes). I limped along using backup to iTunes, turning off and back on individual apps in iCloud (brought back the reminders), or restarting the iPhone altogether (would make the calendar reasonably usable for several some time). I tried to sign out of iCloud on the iPhone, but every time I tried it took forever & never completed. I went through a couple of upgrades, hoping for a fix - to no avail. I spent a few hours with Apple support on the phone, but they were at a loss, suggesting a restore as a new phone - which I wanted to avoid for obvious reasons.

Finally yesterday I saw yet another OS update & thought I'd give it a try. I backed up the phone to iTunes, and started the update, which went through all the stages (on iTunes it actually said it's done, just wait for the iPhone to reboot), but the last reset of the iPhone would never recover, and stay over 12 hours with black screen & a white apple & a status bar almost complete. Restore from backup ends up the same way. I can successfully restore as new, and the barren phone works fine, but any attempt to restore from backup ends up the same way: white apple & almost complete white progress bar on a black background. I could probably live with the manual piecing of all missing apps & settings, but unfortunately I did not back up my photos - I do not use Photo Library and only the last 30 days of Photo Stream show up. I am also missing all Messages history and probably more stuff...

It seems like something in the backup is not letting the restore complete successfully. Probably the same thing that was breaking my iCloud backups and upsetting reminders & calendars. What other steps can I try to restore the backup? And if that's a lost cause - is there a way to selectively recover objects from a backup - photos, Messages history?...

Any help will be very much appreciated!