Error 0xe00002ca black screen after login, Mountain Lion on Macbook pro 13 inch early 2011 SSD 256 GB

Question

After entering my pw at the login screen, I get a black screen
This started after battery got fully drained accidentally.
It has happened in the past after battery gets fully drained accidentally, and I have to reboot several times until it boots properly.
This time I tried rebooting several times but it never booted properly
Tried booting into Safe Mode, by restart Mac while holding down theShift key past the chime, until I saw the Apple logo and spinning gear. I did NOT see the spinning gear change into a solid progress bar. I also did NOT see OS X login window with the words “Safe boot” in bright red.
I Tried booting Mac’s recovery system, by restarting Mac and holding down Command + R during boot. I ran Disk Utility and theVerify Disk option  did NOT find any errors. Interestingly only 16 mb was found to be available on the 250 GB disk. Note: I have filevault2 encryption enabled
Please find below the output of verbose mode.  (Command-V)
com.apple.launchd.     1 com.apple.launchd.
** /dev/rdisk1 (NO WRITE)
** Root file system
   Executing fsck_hfs (version diskdev_cmds-557.3.1~5).
** /dev/rdisk1
        Using cacheBlockSize=32k cacheTotalBlock=98304 cachesSize=3145728k.
** Root file system
   Executing fsck_hfs (version diskdev_cmds-557.3.1~5).
** Checking Journaled HFS Plusvolume
    The volume name is Macintosh HD
** Checking extents overflow file.
** Checking catalog file.
CoreStorageGroup::completeIOrequest - error 0xe00002ca detected for LVG "Macintosh HD"(F2296C53..........), pv 22837757 .........., near LV byte offset = 1823211520
disk1: I/O error.
AppleUSBMultitouchDriver ::checkStatus - received Status Packet, Payload 2: device was reinitialized
** The volume Macintosh HD could not be verified completely.
Kext loading now disabled
Kext unloading now disabled
Kext autounloading now disabled
Kernel requests now disabled
com.apple.launchd.     1 com.apple.launchd.
N 20 21:05:51 2014
C
ocontinuing
m.apple.launchd.     1 com.apple.launchctl.system
done
CPU halted
I do have time machine backup.
Do I need to erase the harddisk and restore from time machine, just reinstall OS Mountain lion from internet, get a new hard disk, or do something else.
I don't know if I need to boot this Macbook from an external disk/drive and copy data or fix some minor error
PLEASE HELP !
I would highly appreciate it.

Answer 1

Try booting into the Safe Mode using your normal account.  Disconnect all peripherals except those needed for the test. Shut down the computer and then power it back up after waiting 10 seconds. Immediately after hearing the startup chime, hold down the shift key and continue to hold it until the gray Apple icon and a progress bar appear. The boot up is significantly slower than normal. This will reset some caches, forces a directory check, and disables all startup and login items, among other things. When you reboot normally, the initial reboot may be slower than normal. If the system operates normally, there may be 3rd party applications which are causing a problem. Try deleting/disabling the third party applications after a restart by using the application un-installer. For each disable/delete, you will need to restart if you don't do them all at once.
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Safe Mode