Given existing low-orbit and geo-synchronous satellites, would it be possible to build a device that could send a signal to a satellite and then have that signal received on another device? Essentially I'm wondering whether it would be possible to reverse-engineer satellite phones (without just dismantling an existing one) to send/receive signals, and if so, what would be the steps involved? Both in terms of hardware as well as any additional steps needed (e.g. figuring out the frequency that existing satellites are listening on, any encryption/decryption that would have to happen, etc).
As a radio matter, yes. Amateur radio operators have built and contacted their own satellites, had conversations with Astronauts and Cosmonauts who were licensed Amateurs orbiting on the Shuttle, MIR, and ISS, and bounced signals off the moon, which is not only far beyond even geosynchronous orbit, but a fairly lossy reflector compared to an active satellite transponder. And done all of these with home built equipment as well as commercially purchased gear.
As an access code matter, probably not, at least in the intent of the satellite owners. Commercial satellites, especially those with end user terminals sold to the public are generally going to be somewhat locked down. Someone might be able to figure out how to emulate an approved terminal like a satellite phone (especially an aging scheme where flaws might have since been discovered), but it's certainly not the design intent to support DIY efforts.
As a legal matter, almost certainly not. Most commercial radio services are limited to
type accepted equipment (or else that signed off on by a licensed professional technician) which excludes that home built by an individual. Home building amateur radio gear is permitted, but as with all operations under the Amateur Radio Service, that can only be used on frequencies where amateur radio operations are permitted, which is not where commercial satellites would be. And even if there is a case where a commercial satellite allocation and an amateur radio band share the same frequency range, such cross-service operation would not be permitted. These rules will be fairly consistent from jurisdiction to jurisdiction, differing slightly in allowed frequencies but not in general intent.