In order for it to make the three months travel time does it still have to wait for a transfer window or can it go at any time of the year?
A skyhook is simply about providing a free boost to the spacecraft, it isn't a magical teleportation device. All the normal realities of interplanetary transit apply. Launch windows are simply the time when the energy requirements are at their lowest, the energy requirements go up substantially outside the optimal times and there's a period where there's no direct path possible because it would go too close to the sun.
Skyhooks are substantially limited in their maximum possible ejection velocity, you're not going to get to deviate all that much from the standard launch window with the energy it will provide.
Note that elevators as opposed to skyhooks have no theoretical maximum velocity, they're only limited by the strength of the elevator cable and competing astronomical bodies. (For example, a cable on Earth must stay below lunar altitude, thus limiting it to 356,000km and thus limiting the tip speed to 25.8 km/sec and the max ejection velocity to slightly less than this. That's not enough for the highest energy trajectories to Mars.)
Note, also, that you have to stop upon arrival. The faster you make the trip the faster you're going to arrive, the more heat shield you'll need and the more acceleration you'll take. (Considering that maximum-energy for a launch from an elevator on Earth you need the entire diameter of Mars to keep the deceleration to --and that's going to require curving around the planet to stay in the atmosphere while you're slowing. I don't think it's possible.)
In practice a skyhook gives you a slightly wider launch window, that's all.