Solid advice depending on the circuitry, or could be similar to splitter just inside the chassis after the input selector. Definitely worth checking out. 
Very true. It could be. Any passive input splitter will present two paralleled loads to the DAC’s output, and is also an easy way to earth-loops too.
In this case, one of those runs should at least have a tiny audio-transformer in-line to isolate one feed from the other. Very cheap. A D.I. box even, with ground lift. The transformer might rob 2dB of signal, but it will allow the DAC to produce expected levels.
If he connects his headphone amp to an integrated amp, why add another component to the chain if he doesn't need to? Since there's no cable, will these RCA splitters drastically lower the impedance on the DAC?
Yes. You’ll still have paralleled loads of combined lower impedance.
And you’re now putting a physical load on that poor circuit-board mounted RCA socket.
Something like this at least, is needed for each channel, to achieve proper separation from dual-loading…
