I'm no hardware expert , but looking at your specs - it would be astonishing if your pc can't support x64 o/s.
The ram thing is normal as you are currently running 32 bit o/s.
This is one of my preferred install methods - see if it works for you.
You currently have 7 x86 installed on partition C ? You can do this:
d/l this .zip , rt click Properties>Unblock, Apply, ok. imagex.zip
extract the .zip and copy imagex into your C:\Windows\System32 folder.
Put the x64 dvd in the drive - open to view files - in the sources folder - you will find install.wim. Copy and paste install.wim onto the C drive.
Make sure you have another ntfs primary partition ready made to install 7 x64 - e.g. D partition. ( make it smaller than your current C partition - 16gb is recommended - you could get away with less)
Open an elevated command and type:
imagex /apply c:\install.wim 2 d:\
it will take about 6 mins.
type: d:\windows\system32\bcdboot d:\windows
then press enter. close cmd prompt.
Restart the pc and select windows 7 from the boot menu. Should be finished in 5 mins or so.