nmigen VS xvcd

You can either decide to learn VHDL/Verilog, or use something like nmigen. I recommend learning either Verilog or VHDL anyway, so you can at least read and understand existing designs, but I personally use nmigen.

Alternatively, you can just use Vivado to build the bitstream and then use alternative tools like https://github.com/trabucayre/openFPGALoader and http://xc3sprog.sourceforge.net/ to upload the bitstream to your FPGA. This is what I do since I use nmigen myself.

Yes, any board with the FTDI FT232H works as a JTAG programmer (I actually have the one you linked). You may probably need some jumper wires that have the Xilinx-specific 0.2mm pitch, though, rather than the more convential 0.254mm or 0.1" pitch, if your FPGA board has the actual Xilinx JTAG header. To get it to work inside Vivado, you can use Xilinx Virtual Cable, which is a TCP/IP protocol to act as a JTAG cable (see https://github.com/Xilinx/XilinxVirtualCable and https://github.com/tmbinc/xvcd). The idea is that you basically have a TCP/IP daemon that speaks XVC and relies on libftdi or libusb to communicate with Vivado through XVC and do the actual JTAG programming.