I would like to create a desktop app which can send / receive transactions on the Ethereum network. But that would be only one feature of the app. It would have several other features. An example would be the Brave browser, which will use Ether to pay content providers, but is mostly a desktop app.
Is this possible? If so how does it work? Can a desktop app join Ethereum via API calls, without maintaining a full copy of the blockchain?
I think your best bet is to use Metamask with Electron.
Metamask provide you the opportunity of connecting to the ethereum network without running a full node. You may refer here,
MetaMask takes things one step further, though, as this protocol removes the need to run a full Ethereum node as well. Bringing additional convenience to the table, combined with fewer technical requirements to access dApps, is a massive breakthrough for the Ethereum ecosystem as a whole.
And to build your core desktop app you can use electron that allows you to build cross platform apps in a web-like environment using HTML, CSS, JavaScript etc. You may refer here.
Think of Electron as Cordova for desktops. It lets you build cross-platform desktop apps using JavaScript, HTML, and CSS. Where you need to tap into native device capabilities, such as file system, Bluetooth, or USB, you can use node packages. As you might already know, node packages can call native code. This means that electron-based desktop apps can do anything that native apps can do.
All what you have to do is to combine these feature of metamask and electron. I think this blog post might be useful to you in this.
If you want to be a peer in the Ethereum network, you are probably looking for Light Clients. Try Parity (PIP):
parity --light
geth --light
This only maintains a minimal overlay of block headers (a couple of MB in size) and queries other nodes in the network for everything not stored locally via pip or les.