Yes, Rx and Tx should be reversed.
Try different handshake settings in your port. I am using DTR/DSR, but the Asymtek can be set to different modes. You can't reset all personality parameters with one order, but some of them can be set by sending <ESC>.R where <ESC> is ascii 27. It sets DTR...
The RS232 cable for the Asymtek 402 is a standard RS232c cable from the time, when DB25 connectors were used. If you can find a conversion connector DB25-DB9, you can use a normal RS232c cable with DB9 connectors. Only pins 1 .. 7 and pin 20 are connected in the Asymtek.
You don't need a 3/2 valve with external pilot port, since you would connect pressure to the normal pressure port (1), from where supply pilot pressure is taken the normal internal way. Connect port 2 to the cylinder and the normal echaust port (3) to vacuum.
It also sounds a bit strange to me...
You have to turn the spring in the z-head balancing most of its weight. If you want the y-arrow keys to act in the logical way, you need to change the personality parameter no. 5, "PE5,1;". This is it as far as I remember. I did it several years ago.
