Is there a way to take a mesh, cut it with seams or sharp edges and flatten it, so that the faces stay the same shape and size?
Making paper models is one aspect, where you could use that technique.
No, I'm afraid SZ0Return is not going to do it properly.
And misusing the UV-unwrap is only a last option.
Note, while it's an option to revert to Blender 2.49 which had a script to do this, I'd like a solution that I can use inside the current workspace.
There is an exporter for paper models, it even has a Blender Wiki Page with in depth usage guidelines.
For more information and plenty of examples of successful use cases you can read the BlenderArtists thread by the developer of the script.
Add seams like you would for any regular texturing job, experiment with simple geometry first. Here is an icosphere.
For Blender 2.8+ the paper model addon is available at GitHub
Quoting the manual at http://addam.github.io/Export-Paper-Model-from-Blender/
Install
The script is included in Blender 2.80. Just enable Import-Export: Export Paper Model in Blender Preferences → Addons.
If you need something special and know what you’re doing, you may download and install a recent version. There is also another version for Blender 2.79.
How to Use
An optional step before start is to mark edges where the model will be cut. From the Properties editor, call the Unfold operator on the selected model. The edges will get marked as Edit mode Seams, you can edit them in any way you like. The script will use the seams you mark and mark some more as necessary.
All the show happens when you choose File → Export → Paper Model. A File Browser appears, which also allows you to change the export settings in its bottom-left corner.
Then (if everything goes well), the script saves the file without talking. In the settings, if you switch No Texture to something else, Cycles' Bake functionality will be exploited to render the images. That may take some time.
The file format is PDF and should be ready to print. If you want to edit the net by hand, you may prefer exporting to SVG and using a vector editor such as Inkscape.
Advanced Usage
A humble panel is in the Properties panel in Object Data section. Currently it gives you an overview of how the net will be split into parts (aka. islands). It can show one of them directly in the 3D view, highlighted in orange. You can edit the name of each island and give it a three-letter abbreviation that will be used on the sticking tabs.
