Is it possible to run both bitcoind and bitcoin-qt as backend and frontend?
No, you cannot do that. Bitcoin-qt is not just a frontend GUI or wallet handling software. It is bitcoind with a GUI. It does all of the same functions as bitcoind as it is built from the same sources. bitcoind also has the same wallet handling components as bitcoin-qt. bitcoin-qt has the same node, consensus, networking, etc. handling code as bitcoind.