It doesn't matter what the tuning is on your guitar cause the synth tracks the string note that's playing. However, if you put an E string on all six strings, you'd still have to tune each string down to a lower pitch. In other words, if you put a high E on your low E string and tune it down, the synth is going to have even more problems tracking it that just using a low E string.
As far as the low strings go, you kind of get used to playing slightly ahead of the beat to compensate for the slight delay. Although when recording midi, you just quantize everything.

One thing you can do is just transpose all your midi notes down by an octave. That way if you're playing bass, you can do it on the higher strings and it tracks faster.

As far as tuning goes too, at least with the GR800, you have to be perfectly in tune for the note you're playing or you get glitches. You can tune the guitar to any tuning you like, but if the note is slightly sharp or flat, the synth has trouble figuring out which note you want to play.

One thing I'm not sure about with the new synths is if you can turn off chromatic playing. My GR800 has built in sounds and allows you to do bends and so forth but when you use the midi output, you have to play chromatically (ie, if you try to bend upwards, it plays each note individually).
I also don't know if the new synths have a 'hold' function.
With the GR800's internal sounds, when you play a chord, you can press the 'hold' pedal with your foot and it sustains the chord until you let it go.
However, again..it doesn't work on midi sounds. So if you're playing strings, they'll only sustain for as long as the strings are vibrating.
If you're recording, you have to go back in later and extend all the notes.
With Cubase it's really easy cause all you have to do is tell the program to Quantize Lengths or Ends and it fixes it. However in my older version of Cubase I had to go in and extend each note by hand in the editor. (took forever :)