Yes. It will matter as much as it will going into an amp. The tone coming out of the guitar is the tone that either an amp or an amp sim works with. But as they say, "the tone is in your fingers." So for now, do the best you can with what you have.

Direct into cubase through what a mixer?? I would use a pre-amp, or even a pedal with an input impedance designed for guitar. I use a Sans Amp (Tech 21) probably has a FET in there (simulates a tube response) sounds great direct.

We are also now going DI out of our Traynor tube amp into a mixer/PA with a subwoofer. But what I am thinking of doing is mic'ing the Traynor amp speaker, the PA speaker and the sub for a tri-amp/channel sort of setup. Reason being I believe a guitarist will adjust his(her) playing style to suit the sound.

A crappy guitar recorded direct may not be much fun. I agree with the Tech21 suggestion. They fantastic products which aren't terribly expensive.

Since this is Direct and into guitar rig, does the quality of guitar matter much?

I think you probably already know the answer to that question.

Your recordings will only ever be as good as your source.

If the pickups in the crappy guitar are substandard, noisy... if the intonation is wacked.... any fault the guitar has will end up on track.

To what extent will be determined by how bad the guitar sounds to begin with.

Now, the question is.... do you think it matters much?

It all depends on what your expectations are.

The worst thing that you can do is to turn a mix session into a tedious repair session.

There's a huge difference, and when dealing with tracks that are well recorded to begin with, this can produce great results....some songs, if recorded well enough, can even pretty much "mix themselves"....

While the other - the repair session - generally ends up as hours of hair pulling, endless tweaking and fiddling, with a final result of "yeah, I guess that'll have to do".

FWIW