This dx-7 suffered from no output volume. The solution involved replacing the relay which comes just upstream of the outputs (both phones and mains). . It's main purpose is to disconnect the outputs during power up and power down, so that there is no annoying "pop".
For testing purposes I removed the relay and connected the outputs with the little jumper wire you see in the photograph; this enabled me to make sure my diagnosis was correct, and that the keyboard worked with the relay bypassed. It then continued to work once I put a new relay in.
A photograph of the mainboard with the relay removed and bypassed is shown below.
Relays seem to be a common point of failure in older gear, especially audio equipment. I have found bad relays in casios as well.... in fact the muting circuits in many keyboards, primarily in place to eliminate the popping sound present when turning the unit on and off... are in general prone to failure and should be inspected. Other newer keyboards use fets to shunt the outputs to ground, and these fets sometimes go bad and cause distortion or clipped volume.
I recently worked on and RD300 which used fets for this purpose.. they had started to fail and caused clipping and distortion on the outputs, and a considerable loss of volume. I removed them and it sounded like a brand new keyboard.
Anyway, I also put a new battery in the DX7, as well as a socket to make future battery replacement easier (see below).
Loading the presets is possible through midi, but there are a number of steps you have to go to on the DX7 to get it ready to receive midi sysex messages. The info here is useful.