In way to skip the need of trusting a third party and to skip the full download of the blockchain on my PC, I would like to setup a full node syncing an ethereum testnet blockchain on a local server and then connect my PC to this server, using a geth instance.
Is that possible in some way?
How can I do that?
You can just connect to your local geth server via rpc
geth attach http://host:port
There you can read about geth's rpc apis
You can start rpc listener with this command on your geth server after or during syncing
admin.startRPC(host, port, cors, apis)
As an alternative you can just use light sync on your main pc so you won't need to download full blockchain
geth --light
But it depends on what you gonna do with it