Any game update (retail or debug) as long as it is valid and greater version that the disc eboot will run on a DEX console. That means you can take the latest game update for retail, extract it to a directory rebuild the package using psn_package_npdrm, install the package on the DEX console and it should boot the game. You can also install a real debug update it will boot.
On an real DEX you can use your retail PSN account the same as a normal CEX machine. And yes that means that trophies will sync. I can't say if this works with a system converted to a DEX. That entirely depends if Sony keeps a white list of valid machines that sign into PSN/SEN.