Well, I have a similar setup, I think.
I placed PLCC44 and PLCC84 sockets on test PCBs with 10p/2x5/100mil Altera pinout JTAG connectors. I use the TerAsic USB Blaster clone, available at the TerAsic website. Operates just like the Altera device.
I have programmed EPM7064SLC44-10 parts, and EPM7128SLC84-7N parts using QuartusIIWeb 13.0sp1 tools on both a Win7/64b/SP1 (now retired) and Win10/64b/v1909 (my current environment).
I don't have EPM7xxxA series parts but I believe they should program just like the EPM7xxxS series parts.
I attached PDFs of the schematic for my PLCC44 test card, and the PCB layout, for reference.
So what version of Quartus are you using?