FTM 2012 will open any FTM file back to version 5.
It will create a new file with the new file structure and move media to a dedicated folder. I believe that has been true for all the versions 2008 and later.
FTM 2012 runs fine on 64bit Windows 7.
The advise to keep a good copy of the old file in a safe location is good advise. It should not be needed, but better safe than sorry.
Since you have a laptop that is still able to run version 2006, you will be able to compare all your data and identify anything unusual. You should be in great shape.