Diablo II runs just fine in Windows 7. I'm running on Windows 7 64-bit. Just be sure you download the latest patches and then set the icon's compatibility mode to XP. After that, it should run just fine.
BTW, I have the Diablo II Battle Chest version which includes the LOD expansion. Again, it all installed and runs just fine on my machine. Didn't have to dual-install.
Now, for Diablo 1, I just set up a virtual machine using Vmware (which does 3d acceleration which is not necessary for Diablo 1) and it runs fine there.