How do I set external drive as default storage for Mac Studio without booting from It?

Moving, storing and accessing files or libraries to an external hard drive is fairly straight forward.

Create new folders on the external drive, then move your documents and files to them.

App's and Users should be kept on the Mac's startup drive.

If you are looking for more space, move your Photos, Music, Movie Libraries to an external hard drive.

Move your Photos library to save space on your Mac - Apple Support

Change where your music files are stored on Mac - Apple Support

Move your iMovie for Mac library - Apple Support

Just keep in mind, that external drive should be backed up to a second external drive to protect your files and libraries.

The Mac will not treat the external drive as an extension of the internal one. If you want to move things like Parallels virtual machine images to the external drive, you must move them there.

I believe that for most types of iCloud synchronization, any local storage that's needed will come from your startup drive. In the specific case of iCloud Photos, it might be possible to move your system Photos Library (the one that synchronizes with iCloud) from your startup drive to another drive. Ideally one that is going to be mounted all/most of the time, so that the Mac can carry out synchronization tasks in the background.