Yeah that's normal. You just don't want it turned up too high so that it affects the sustain. The other option is turning the stab control all the way up and the almost all the way up, but that doesn't sound quite the same to me.
I haven't tried the file myself yet, only just had a quick look at it. It's definitely a tricky one to get a decent sound for, and there are different ways you can approach it. For what it's worth, Matt seems to have recorded it with the 007 which does have a P90 (single coil) in the bridge. If that's the case, it would likely be pretty noisy with a fuzz pedal on high gain settings. But as far as I know, it's that same setup where he's running it direct, and has the input gain cranked up so it's clipping there (while having a reasonable setting on the fuzz pedal)
And maybe - I'll have to pay closer attention to what he's playing in the live videos, but it sort of sounded like there were two sounds at once. I did find the same about the 5th fret compared to 12th fret though.
Here's what the DT file sounds like at the minute. No video this time, but better sound quality anyway. The expression automation is exactly the same as in the picture, but I corrected some of the program changes (mainly the ones for bypassing, so it doesn't make much of a difference I guess.) This is with the low A notes played on the 12th fret, using settings from both sides of the pedal.
The nut strikes will sound a bit off in this one. The guitar has a 3+3 headstock, so I just hit the open strings instead. Should still give you an idea though. This was just with amp distortion and some post EQ to brighten it up slightly.
One downside of the DT that I think I mentioned was that it doesn't seem to track MIDI well. You can hear that on the wider/slower pitch bends (the -2 octave ones).
As for the lag issue, go to the external instrument plugin and make sure the auto compensate latency box is ticked. I know mine has a significant lag if it isn't.