Yes that would work.
Lazarus only stores info in its config folder. And what is in the install folder. Nothing in the registry (the installer does, so it is listed in installed apps, but not Lazarus itself).
So zipping those folders, saves all you have. You need to put them back in the same place. The config has full paths stored.
If you used the installer to create 2ndary installs (with the config path that you have), then the info about where that config path is, is stored in a file called lazarus.cfg in the same directury as the lazarus.exe (main dir of install).
So that gives you all the info you need to backup/restore/update your install(s).
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About hardcode path:
If you move the folder of any package (e.g. 3rd party package), the IDE will not find it. But if you use menu package > open package to open it, the IDE will remember the new path (rather than you having to edit config files by hand.
If you move the FPC folder, you need to find fpc.cfg inside it, and update it. And under Tools > Options, correct the path to fpc, fpc source and make.