Hi
Since you'll have to use the Xilinx place and route tool, you should be able to create a bitstream that you can download using a xIilinx download cable. Alternatively, you _should_ be able to use the EDA netlist to feed to Xilinx ISE. As I recall, Altiumn wrote their own synthesiser. That must give you an output file compatible with the Xilinx P & R tool. I can't remember what the extension is, I think it is .NGD (netlist generic database) but it will be easy to find out from the docs. I
I confess I'm not entirely sure of what you are asking. Does the Digilent board put the bitstream into a prom of some kind.If so, the Impact tool will create various file formats for a device programmer. I cannot see how you can download your bitstream without some form of jtag cable, unless you can remove the bitstream prom and use a device programmer.
If you could elaborate a bit I can be more helpful. Meanwhile,the intermediate files that are produced to change tools (sch->HDL, HDL-> synthesiser, synthesizer-> P&R, P&R to bitstream, _should be in EDIF formats and exchangeable between tools. There are, for example, numerous third party synthesisers from (e,g.) Synopsis, Aldec and many more. They all produce a generic EDIF file as only the manufacturers can do the Place and Route stage.
One more thing, Xilinx are pretty helpful to students, you can get Webpacks for all ISE versions for nothing. The AltiumFPGA tools were never much use. I think they were hopelessly over ambitious trying to make a universal FPGA development system. Besides, Using schematic as input is not my preffered choice. It's cheating, and if you look at the HDL output it can be a huge unreadable mess.Learn VHDL or Verilog. You'll be doing that when you get a job!
Frankly I don't see how you can do this without a jtag cable of come sort.