1. What worked best for me before I bought a DT25 and started using the L6 Link was to use the HD500's right/mono (or left/mono) output straight into the effects return on the amp. Many people here have also had great results from going straight into the sound board, but it seems that your sound guy doesn't know how to EQ your tone the way you want it (the only solution there is to force him to dial back the mids and highs to where YOU want them to be, not where HE wants them to be).
2. No, don't use the headphone outs - use the HD500's left/right output jacks, that's what they're for. Although, frankly, I've found that stereo output usually doesn't work all that well in live situations; after all, you want the audience on the left side of the venue to hear the same mix as the audience on the right side (and middle). IMO, I'd just use mono output. But if you're going to be recorded, you can use the XLR left/right outputs to the recording board.
3. If the HD500 is going directly into the amp's guitar input, set the line/amp switch to amp. If it's going directly into the effects return (or, on some amps, the Power Amp In jack) set the switch to line. And, of course, you can always experiment and set it to whatever you think sounds best.
4. You can use the recommendations in the HD500's manual (see below) as a starting point, but many people end up setting it differently from the "recommended" setting. If the HD500 output is going into the power amp or effects return of a combo or head amp, then you'd probably use Combo Power Amp or Stack Power Amp as the output mode (and set the line/amp switch to line). But there are no set rules - set it to whatever sounds best to you.
- Studio/Direct: Typically the best option when connecting "Direct" to a mixing console or recording device. This signal includes "Studio" Cab Models, Mic Model and an "AIR" convolution to best emulate a micâ€™ed amp within a room.
- Combo and Stack settings: Optimizes your tone for connecting to an external amplifier. The signal includes "Live" Cab Models,with no Mic Model or "AIR" convolution, with a special EQ curve added for each.
- Combo Front & Stack Front: For connecting into the front input of a typical combo amp or amp head + external cab, respectively.
But, seriously, take a look at the Line 6 documentation - it will make things a lot easier for you.
Good luck with the gig!