Finding lost partitions
As the gpt command reveals you have two recovery partitions (and a lot of unallocated disk space after partition 4):


partition 3 with a wrong size (1269544 blocks) and a correct partition type (426F6F74-0000-11AA-AA11-00306543ECAC)
partition 4 with the correct size (1269536 blocks) and a wrong partition type &amp; visibility


The second Recovery HD (slightly visible below the Macintosh HD in your screenshot) is blocking the expansion of the volume Macintosh HD. The easiest procedure to solve your problem is to boot to Internet Recovery Mode and completely reformat the internal drive and reinstall OS X:

According to this Apple support document your MacBook Pro (Mid 2010)* can be upgraded to use Internet Recovery after applying a firmware upgrade. Since the firmware upgrade was published in 2012 I assume that it was applied already.

So to answer the first part of your first question: After booting to Internet Recovery Mode you may install Mac OS X Lion. In Internet Recovery Mode usually the Mac OS X the Mac originally was shipped with will be installed. For Snow Leopard Macs like your MBP Mid 2010 Mac OS X Lion will be installed.

To initiate the Internet Recovery Mode restart your Mac and press alt cmd R.
The prerequisites for an Internet Recovery are listed here: Recovery/Internet Recovery.
In a few words you need the latest firmware update installed, either ethernet or WLAN (WPA/WPA2) and a router with DHCP activated for a successful internet recovery.
On a 50 Mbps-line it takes about 4 min (presenting a small animated globe) to boot into the Recovery Netboot image which is loaded from an Apple server.

After booting successfully open Disk Utility and repartition your disk to 1 JHFS+ partition with a GUID partition table. Quit Disk Utility and start 'Reinstall OS X'. Depending on your internet connection the reinstall will take some time.

The second part of question 1 as well as the questions 2 and 3 are obsolete after following the above instructions and successfully reinstalling Mac OS X Lion.

*: A MacBook Pro (Late 2010) never was produced/sold, so I assume it's a MacBook Pro (Mid 2010)