My iMac started running slow now won't start at all

Question

Two days ago, my Mid 2011 iMac 27-inch, OSX Lion 10.7.5 started running very slow.  Had trouble opening and closing apps.  Then after restarting, it took an extraordinary amount of time to start.  Had to shut down by holding in Power button.  Restart failed a few times, then I was finally able to get to desktop but unable to open any apps.  After many tries and waiting about 30 minutes between each keystroke for something to happen, I got to Disk Utility.
Under the First Aid tab, I ran Verify disk and got "The volume Macintosh HD appears to be Ok."  Repair Disk was not an active button.  At one point I got to the Force Quit menu under the Apple and force quitted Safari, iPhoto, and one other app (maybe Disk Utility) but then still was not able to Shut Down the computer.
Some background info that may or not be important,  I have been unable to Empty Trash or Secure Empty Trash for a long time.  Most of my hard drive filled capacity (60%) are photos and videos.  After running for what seems a very long time, I'd get an error message about not having the right permissions to delete some files in Trash. 
Back to Disk Utility, I tried to run Repair Permissions and Verify Disk Permissions, and both just ran forever (hours) until I stopped them.  After 6 hours of reading possible solutions in this community but being able to perform none of them, I unplugged the computer because I couldn't get to Shut Down.
This morning, I plugged it back in and upon hitting Power, it took about 10 minutes to get the User icons.  I clicked on mine and got a gray screen with the spinning circle.  That was about 30 minutes ago and nothing has changed. 
I welcome to all offers of help.  My concern is that I was in the middle of copying folders upon folders of photos and videos to an external hard drive for the past several weeks and still have not completed that project.  I look forward to a solution that would not result in the loss of photos an videos.

Answer 1

When I selected Disk Utility, the Repair Disk button was active.  It ran for a few minutes then read, "The Volume Macintosh HD appears to be Ok."  Followed by "Volume repair complete" and "Updating boot support partitions for the volume as required."  I have no idea what that last statement means. 
I just ran "Verify Disk" and  "The Volume Macintosh HD appears to be Ok" was the result.
I also ran "Verify Disk Permissions" which wouldn't run yesterday but did just now.  Ended with "Permissions verifications complete."
I ran Repair Disk Permissions, "Permissions Repair Complete."
What should be my next step?