Question I have set the input level pretty conservatively at around -6 dB, and if all effects are bypassed, everything is fine. Then I've been stepping through the presets, and in a lot of them the overload LED start flashing even though there is no perceived increase in volume. Why 

The Overload LED will flash each time there is an overload in the DSP path - So not only at the input.  Some effect blocks with high settings (Specially EQ, COMP and some Filters) can boost and overload the signal even if the input is OK.  The solutions are to back up a bit the input level and / or re-adjust block settings.