partition error. couldn't unmount disk

I can't see any of your images. But you should be able to re-install OS X by booting up holding cmd-r, into your Recovery Partition?

Yes, I can see your images now from 3 of the links you provided. The 4th is dead.

But from what I see it failed when you attempted to create a 500GB partition on a 500GB drive?

In any event, when happens if you shutdown and startup holding the cmd-r keys?

Yes, your diskutil output is a mess.

Again,when happens if you shutdown and startup holding the cmd-r keys?

Yes.

When your internet recovery is done, you may or may not still have those partitions on your disk. The first thing I'd suggest is to make a clone of your OS X volume to an external hard drive. Then, if you need to you can erase and repartition your internal and just clone the system back.

It's really not adviseable to partition your main drive.

Yes, from the Apple server.

My macbook air (OS x lion ) hard disk got crashed. now it starts with a grey screen and MAC OS X utilities. in the disk utility, the hard disk icon became inactive. when i erased the disk it became active again but i am still not able to partition it. it showed -- "Partition Error". "Couldn't unmount disk". now the disk icon became inactive again. when i click on mount it becomes active again but partioning is not done after many attempts.

another problem: - when it I click to proceed with the installation (still unpartitioned) it verifies my macbook's eligibility and says that the machine information is not with apple. below are the screenshots.

is there any internet issue, i got connected on my institute's wi-fi (excellent speed- 54mbps)

Make sure you are using Internet Recovery by holding down Command-Option-R (http://support.apple.com/kb/ht4718), this should prompt you for a full reinstall. When you get the message "Your eligibility will be verified" make sure you are connected to the internet. If you still can't install from here then you need to access the Mac App Store and re-download Lion from there.

If you are unable to do any of the above you have a drive issue and need to take it to the Apple Store.

Good luck!