My future task for a DV7 with the same problem...
I have looked at dual-booting a HP Pavilion DV7 with one HDD and the same four partition layout. For the first, Google told me that if I removed the System partition, Windows 7 would not boot.
I am reluctant to remove the Recovery partition, although I can delete it and buy a set of Windows 7 recovery disks for this machine. In addition, the contents of the HP_Tools partition seems to be available on the HP support site for this DV7 model, and the HP_Tools partition seems the most expendable.
There seem to be two choices, and when I looked at the program contents of HP_Tools, they didn't seem that necessary, and removing them, perhaps after making a clone copy of HP_Tools, is what I intend to do.
After removing whichever of the Win partitions you choose, use the Win 7 partitioning tool to shrink C, the only partition of any size. Defragment first, and if you use a defragmenter such as the one from Auslogics, you will be able to see those Win 7 files that cannot be moved. They limit the amount by which Win 7 can be reduced. Using the Win 7 tool should retain Win 7s ability to boot.
I will use an extended partition for my linux partitions, making sure that the linux boot-loader is installed there. The problem that I expect to have is determining where to point GRUB to chain-load Win 7. I may have to edit GRUB to point to the System partition, perhaps (hd0,0), if that is where to find the Win 7 boot-loader, and hope all goes well.
In the opposite case, these are useful.
I will be using a copy of the PartedMagic live-cd to give me an overview of the partitions and their boundaries before I start to make any changes.