Inclination test
 
I tried running this overnight with a 65k timestep, and recorded the first million year's worth of evolution, and I got no noticeable change in eccentricity of the inner gas giant (beyond the usual small scale cycles), and a very very slight increase in its inclination. I just got some more data points for around the 10 million year mark and the inclination of the gas giant has gone from 0.00 (at the start) to 0.02 (at 1 million years) to 0.24 degrees (at 11 million years) - the eccentricity hasn't changed though.

I've heard of this before.  It's related to the Kozai mechanism.  I have not yet simulated it for binary stars, but on the "Simulations" link there is an article called "Kozai Mechanism".  It shows 2 examples with Gravity Simulator.  The first places the Moon in a circular polar orbit around Earth.  All seems fine for a few orbits, but then eccentricity is pumped into the Moon's orbit.  Within a decade, its perigee is below the surface of the Earth  
 
The next example doesn't actually simulate the effect, but shows what is theorized to be caused by the effect.  The jovian system of moons contains no moons with inclinations above about 60 degrees.  The reasoning...  If they existed, the Kozai mechanism would pump eccentricity into them until their perijoves were in the vicinity of the Galilean Moons.  Then their days are seriously numbered (think 5-planet system   ).
 
Comparing a binary star system to the Earth / Moon simulation, the two stars are analgous to the Sun / Earth, and the planet is analgous to the Moon.  There's no reason Gravity Simulator shouldn't be able to handle this, although the time scale required to see results may be very long.
 
Notice in this simulation that it takes a while for the action to begin.  The Moon orbits the Earth normally for several orbits before you notice any significant eccentricity develop.  But once it does, the rate accelerates.

Right, but the problem I have with this is the idea that an inlcined brown dwarf 500 AU from the star can screw around with the eccentricities of a planet orbiting really close to the star. That sounds really counter-intuitive to me.

It makes it annoying too, when you're trying to write a worldbuilding program!
 
That said, it only holds if the BD or companion star is on a very inclined orbit (45° or more) relative to the planet/ecliptic. I'm not entirely convinced that's likely - wouldn't they have formed in the same plane because the companions form from the same disk of material as the primary and planets?

Concerning the last question : I think the Kozai comes only in action if the inclination is big  related to the plane of motion of the mother planet . The oriontation of spin around its axe doesn't matter ....

25 million years into it, and the inclination of the gg has gone from 0.26 to 0.56 in the 14 million years or so since this morning. Eccentricity is still cycling in exactly the same way as it did before though, at 0.6 +/- 0.05 - that hasn't changed at all each time I've looked at it.
 
So... over time, inclination is increasing from zero, but eccentricity is just staying stable (on average).  
 
The inclined BD seems to be the same as it ever was.

Check out how sensitive the Kozai Mechanism is to distance.  I ran a simulation similar to the one on the website where the Moon was placed in polar orbit around Earth.  But in this sim, I had 5 Earth systems:
 
Earth 1: 1 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
Earth 2: 1.5 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
Earth 3: 2 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
Earth 4: 2.5 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
Earth 5: 3 AU
Moon : 384,000 km
Inc: 90 deg
Ecc: 0
 
And the data:
 

 
The Moons of Earth 2-5 are all crammed together on the bottom, barely visible with the scale high enough to show Earth 1's Moon full range.  I did not give the Earth's any size in this simulation to avoid collisions.  So it is interesting to note that the Kozai Mechanism does not pump ecc>1 into a system.  Once it approaches 1, it backs off, and I'm guessing periodically goes back to close to 0 and repeats.
 
Leaving the data for Earth 1's moon off the chart to do a better job of plotting the moons for Earth 2-5 gives this:
 

Again, the Moon from the closest Earth significantly outpaces the others in its eccentricity gain.
 
I think I'll run a similar sim tommorow, but with Earth SMA values more closely surrounding 1 AU.
 
This was run at a slow time step of 16 (not 16K !), for only about 7 years.
 
Try it yourself:
http://www.orbitsimulator.com/gravity/simulations/kozaitest.gsim