I've never used the original Spartan devices, but that sounds correct about no synthesis in ISE Classics. It only provides place-and-route and downloading. Synplify and Synplify Pro are good 3rd party synthesis tools that support the old Spartan FPGA. When you create an FPGA project in ISE Classics, the Project Navigator tool can automatically call Synplify to compile your HDL. That's convenient.
You have an unfortunate mix of old and new devices. ISE Classics supports the XCS05 but not the XC9536XL. ISE 7 supports the XC9536XL but not the XCS05.
**broken link removed**
You probably need to install both versions of ISE, and also a 3rd party synthesis tool such as Synplify for the XCS05. That's messy!
If you somehow find an original set of ISE Foundation 4.2 disks, I think it included an HDL synthesizer that works with Spartan. It was probably FPGA Express or an early version of XST. However, many folks reported those old version synthesizers weren't as good as Synplify.
I think the JTAG download utility (iMAPCT) in all newer versions of Xilinx ISE will download to all older Xilinx devices such as the XCS05.