You won’t be able to hook everything up they way you’ve described. The RCA outputs of your head unit are too low to power the speakers - they are for connecting to an amplifier only. You will need to connect your speakers to either the speaker-level outputs of your head unit or to a separate amplifier. You can tap into the speaker outputs of your head unit as long as you can find which wires go to which speakers. If necessary, you can trace the wires from the head unit to the speakers to see which wire is what. It’s okay if you can’t tell which is + and - because you can wire them up either way and it should work. If the speaker polarity is reversed the worst that can happen is poor sound quality and then you can reverse the wires to check if it improves.
Since you have the RCA outputs on the head unit, you could easily add an amplifier if you’d like. Basic amps aren’t too expensive and you don’t need anything fancy. They do add a level of complexity to the whole installation so maybe consider professional installation.
You’re right that you need to use the crossovers that came with your speakers and the speakers won’t work properly without them. The “Amp” inputs on the crossovers can either be connected to the speaker outputs of your head unit or to a separate amplifier as described above. It would be a good idea to invest in some new speaker wire since using broken wire is dangerous and can damage your equipment. Any good quality speaker wire will work - it doesn’t have to match the old speaker wire.
Good luck!