Is it possible to back up and restore a desktop to a new version of ArcaOS?
No. eCS offers some kind of migration, but restores not much. It has never worked that well, that you want to prefer that method over the normal re-install.
As Dave wrote, if you have scripts or installers for your additions, you should be able to restore them easily.
Details: Settings are stored in the OS2.INI and some in file object EAs. OS2.INI entries rely on object handles. Object handles are saved in OS2SYS.INI. These handles change with each installation. The EAs are saved in the file system, but a new installation won't respect these old EAs. It simply overwrites file objects on the installation partition. (EAs from objects of other partitions won't be changed.)
To restore ini settings, you have to extract object settings without the dependency on object handles, but e.g. on object ids or file names. For file objects, you have to extract the relevant EAs before they get overwritten. In your case, I guess, these are .ICON and (for folder objects) .ICON1.
Object settings: You could try WPSBKP.EXE and WPSREST.EXE by Henk Kelder. These belong to the WPTools package (WPTOOL32 on Hobbes). Note that only some object settings were restored and that the .CLASSPATH EA is undocumented.
WPS settings: They were saved to OS2.INI and will be overwritten by a new installation. E.g. you can extract XWP's settings from the old installation and restore it to the new one. But the script for it has to be created by yourself.
Object Desktop's object save and restore facility works well with objects, but can't know about all WPS additions how they save their settings in OS2.INI.
Additionally, there are applications that save their settings in OS2.INI or save their own .INI file in the \OS2 directory. (Unfortunately that happens even for non-system applications and can't be fixed in some case, because apps were abandoned.) These settings also have to be extracted in advance.
Applications, which make it right, store their .INI file in their application tree. These reside usually on another than the installation partition and won't get lost. Only the WPS objects get lost, but e.g. in the case of AOO, there exists a script to restore these objects.