Updating OS X El Capitan to macOS High Sierra as a developer
First off, why do you need to update beyond El Capitan? Yes, it will be end of life in a little less than a year, but if your concern is just getting started with the development project, you don't need to update macOS. A project built on Xcode 7.x and El Capitan will work just fine on (High) Sierra.

Putting that aside, yes, the latest Xcode will require the latest version of macOS. If you've done Time Machine backups of the box, then you shouldn't run into any issues with the OS and environment provided the backup is good (which you've indicated). For extra peace of mind, you could copy the project itself to an another disk or just make sure it is up-to-date with the source code repository (it IS in version control, right?!)

A project created with Xcode for El Capitan will (I believe) work without modifications on Xcode for High Sierra. However, you'll need to update the project for it to take advantage of the features of the newer OS / frameworks (e.g. something like Siri integration wouldn't be available without re-targeting the project to a newer framework version). Just be sure that if you're going to move to the newer version that you stay mindful of what minimum OS X / macOS version you want to support; you can easily lock out anything older than High Sierra which might not be your intention. In general, you want to support the minimum OSX version that has the features you need, so perhaps Mavericks (10.9) or Yosemite (10.10) would be a good starting point, but I'd probably just leave it as it was set by the previous developer.