I have this machine, but it is badged as a Faema e98 compact. Same electrical setup with the two rotary switches on the front base.
The only part to have failed on my machine has been the brew solenoid. If it was me, on your machine I would remove and replace the entire solenoid, then maybe descale the original and hang onto it as a spare. Given the amount of scale at the joints of your pipework Im guessing that solenoid is probably gummed up inside. You mentioned you bought this on Craigslist, Im guessing you are in the US? Is your machine running on 110V?
Given my machine is running on 240V my setup is possibly different to yours. For what its worth my brew solenoid was a Sirai brand [Sirai Type ZA32A, V220-240 Hz 50] but I replaced it with a Parker. I feel maybe the Parker sticks a little more than the Sirai but it doesn't really effect the functionality and it reminds me to keep up with the back flushing.
As for the pipe joins with the scale and rust, those joins should be just copper and brass components, I'm not sure where the rust would be coming from. Looking at the photos Id say a full rebuild really needs to be undertaken. All the leaking joins need to be cleaned and re seated or they will continue to leak and cause problems down the track.
Good luck with the project. Once it is up and running the machine is a very dependable one and capable of making great coffee.