Issue 1:
Assuming you are talking about the V1 version of the boards, SparkFun currently sells them
Depending on your application, you could also use a Br board to avoid using these connectors directly. Alchitry Br Prototype Element Board
Sourcing these could be a pain and was a big factor in the change to the easier to find connectors on the new V2 boards. Products – Alchitry
You can find these at many retailers like DigiKey
https://www.digikey.com/en/products/detail/hirose-electric-co-ltd/DF40HC-4-0-80DS-0-4V-51/2530298
https://www.digikey.com/en/products/detail/hirose-electric-co-ltd/DF40HC-4-0-50DS-0-4V-51/5132169
https://www.digikey.com/en/products/detail/hirose-electric-co-ltd/DF40C-50DP-0-4V-51/1969486
https://www.digikey.com/en/products/detail/hirose-electric-co-ltd/df40c-80dp-0-4v-51/1969492
Issue 2:
Are you connecting the Xilinx cable to the JTAG pins on the header or trying something else?
The easiest way to use JTAG is with the on-board JTAG bridge. You can program the V1 boards with the “program_ftdi” program included with Vivado. It will then show up in the hardware manager. See No Devices Detected with Vivado Hardware Manager - #5 by alchitry
The new V2 boards come with this pre-done.