MacBook Pro (El Capitan)

When you say you ran the builtin First Aid tool, do you mean you booting into the recovery partition by booting while holding down Command-r, the highlighted Macintosh HD and ran Repair Disk? If not, try that.

Otherwise, pull down the Apple Menu to Force Quit and then try to shutdown. See if any apps show as still being open in the Force Quit window after you have issued the shut down command.

If it still refuses to shut down, try again, but this time make note of the exact time shown on the menubar clock when you issue the shutdown command. Then if it doesn't shut down, force a shut down by holding down the power key. Then after rebooting, open console.app and copy and  the log from the time that you tried to shutdown until the reboot started and paste it back here.

Does Command-Tab show you any applications? I've seen behaviour like this with non-Safari browsers installed on El Capitan. What browsers do you have installed?

I did First Aid through Recovery Disk during bootup.

The laptop also has chrome installed.  Not sure if I should uninstall it.

Does the laptop have any external drives attached, like for running Time Machine?  I've had instances in the past where my Mac would not complete a restart or shutdown until I removed the external drive. What I've gotten in the habit of doing is to eject them before shutting down or restarting.

No harm in uninstalling Chrome and see whether that makes a difference.

Chrome should not be causing a problem.  Open Console.app to look through the logs to see if there's an error message indicating what might be causing the issue.

Try resetting the SMC and PRAM.  These are among the 1st things you should try to help isolate the problems.

STRUNG:  I am attaching the log file.  I had to force shutdown and the time was 9:40 am.  Logged back in at 9:45 am.  I am sure the log has many more info than needed but I am attaching it anyways
All-Messages.log

SERIALBAND:  I did SMC And PRAM reset with no luck of fixing the issue.

I uninstalled Chrome.  No luck

The laptop seemed to have external drive used for Time Machine.  But I have totally disabled it by running the following command:

sudo tmutil disablelocal

Still no luck

STRUNG:  After I issue the command to shut down and most items disappears from desktop (as though going to shut down), It hangs.  

Then I issued the Force Quit command by pressing Command+Option+ESC since I have no menu bar to pull down to force quit.  Once I issued the command, I get a pop-up window with FINDER showing.  My only option there is to relaunch.  I could not quit FINDER.

If I click RELAUNCH, nothing happens.  So I have no choice but FORCE SHUTDOWN by holding down the power button.

OK.  I went into Activity Monitor and killed the process FINDER.  

Since then, I have done 2 successful Shut Down.  But I don't understand what transpired.  

I won't feel comfortable that the issue is fixed unless I have several successful Shut Downs and Reboots.

Your log files show some corruption with certain core services.

It's probably best to boot into Recovery Mode and reinstall the core OS.  It's best to Back Up your data before you do this, but a simple OS Reinstall should not delete user files or Apps.

core services, as in?

EtreCheck is a free, fast, diagnostic designed to help diagnose OSX problems.
Running and posting the results may be of use.
Starting in Safe Mode (SHIFT key down) can help diagnose.
The exact model, OSX version, and RAM installed are basic pieces of info.
Downloading the 10.11.5 Combo update and installing sometimes help.

Issue still continues.  Prevents it from shutting down or rebooting

Do you have any external peripherals, particularly drives attached? If so, try disconnecting them and then see if you can shut down.

If that doesn't solve the problem, please post an EtreCheck report as suggested by Dave Anders:

https://etrecheck.com/#download

No peripherals attached during reboot or shutdown or otherwise.

Here is the report...
System-Report.pdf

Unfortunately, the PDF truncated some lines. Perhaps next time you could simply post the .txt or .rtf file.

In any case there are a few things that look a bit suspicious.

In your earlier console logs there were some Spotlight errors and the Etrecheck shows MDWorker which is the Spotlight indexing process using a lot of memory. It is possible the you have a corrupted Spotlight index and Finder won't quit because Spotlight is trying to index.

You could try disabling Spotlight by copying and pasting the following command into terminal. Hit return, enter your admin password and hit return again:

sudo launchctl unload -w /System/Library/LaunchDaemons/com.apple.metadata.mds.plist

After doing that, see if you can shut down normally.

You can later re-enable Spotlight using this command:

sudo launchctl load -w /System/Library/LaunchDaemons/com.apple.metadata.mds.plist

Turning to other possibilities:

You have something called Digidesign loaded. Is this a driver for some external peripheral? If so, try disconnecting the peripheral before shutting down.

Unless you are using a bluetooth keyboard or mouse, try turning off bluetooth before shutting down.

You also have Dymo extension running. Is that for one external peripheral? If so, disconnect it.

You might try temporarily uninstalling BitDefender and Clean My Mac to see if that makes a difference.

I did.

Performed couple of reboots and shut downs and it worked.  Opened Finder and clicked on few tabs, applications, then closed it and exited Finder.  Tried to do reboot, and it got hung again.

Starting in Safe Mode (SHIFT key down) helps to eliminate third party code.
Creating a Test User (Admin)  and logging in as Test eliminates user file corruption.
Downloading 10.11.5 COMBO updater and installing makes you feel like you're doing something.

Davidanders

I tried your options mentioned above.  Here is the result

Option 1.  Failed to reboot or shutdown
Option 2.  Failed to reboot or shutdown
Option 3.  After installing COMBO, system is supposed to auto restart.  But got hung.

As mentioned above, I created new profile and tried it out.  It got hung at the same point.

Yes, that was going to be my last resort anyways, is to re-install OS but I was hoping to avoid that.

I used command+r on reboot to re-install OSX El Capitan.  Took more than 2 hours but after all said and done, it still failed to reboot or shutdown through normal process.

Something really unusual is taking place.  Not sure what it is.

Take it to an Apple Store I would say.

The hard drive cables can fail, I have replaced five or six. The symptoms can be very mysterious.
Removing the hard drive and plugging it in with an adapter is often the only way to diagnose.
Booting with another drive, with the hard drive inside with the bad cable, is not a reliable test.
iFixit.com sells the cable and has instructions
https://www.google.com/search?q=macbook+pro+hard+drive+cable

Bad sectors can be found with trial of DriveDX

I ended up factory resetting to LION and then upgrading to El Capitan.  This fixed my issues.  Apparently, the laptop must have had corrupt files and especially FINDER.  Reboots and Shutdowns work just fine.

Yes, it was a hassle but after much research online, I found out that there was no other option but to backup all critical user created files, and do factory reset and then upgrade.

Ran tests to make sure the drive was healthy.

Thanks.  I was reluctant to reinstall OS but at the end it seemed to be the only solution