it's a small project
Keep in mind that Electrum "discuss" with its own servers (or nodes), hence you'll probably have to also adapt ElectrumX (or electrum-server or similar) too and the project will no longer be so small.
Also in order for this to work, somebody will have to host those servers...
See
https://github.com/spesmilo/electrumxDoes electrum broadcast the transaction to bitcoin nodes directly?
Nope, it has its server.
Is there a handshake once you broadcast the transaction? I mean, can I simply change the default IP addresses of the litecoin nodes so I can put dogecoin nodes? Is it that simple?
Probably ElectrumX will be a bit different too.
What other things do I have to change excluding the address' prefix?
You'll probably have to distinguish between the networks and avoid LTC/DOGE signal themselves as BTC ElectrumX servers.
Although I am against BCash existence, this time it may prove itself
helpful for you, since it has its own Electrum fork.
https://github.com/Electron-Cash/Electron-Cash