Yes all the intake air passes through both chargers in series, all of the time. And trying to get around that somehow, will not gain you much. But it will surely increase complexity, and introduce some other rather interesting little problems that will likely keep you occupied.
Two stage compression does not mean the compressed air will be twice as hot as many people sometimes expect. It is compressed twice, but each stage only provides roughly half the total compression. The net effect is not going to vary by much, no matter how this compression actually takes place.
By far the more important issue is providing sufficient intercooling AFTER all this violence has been inflicted upon the air, (by whatever means).
If intercooler capacity is made generous, the efficiency of compression becomes hardly significant at all. If you can reduce induction temperatures sufficiently, the engine will be very grateful. And it will happily show it's gratitude by providing a little extra supercharger drive torque, and tolerating a slightly higher turbine intake pressure.
So to a great extent, a less than wonderful supercharger and turbo combination can be offset by fitting the worlds largest intercooler. At the end of the day, what comes out of the flywheel is what matters.
Hoof, use one monster intercooler last thing after the supercharger. Cooling the air before the supercharger is not of any real practical advantage. This is especially true if it means the final intercooler must be made much smaller, which is often the case.
The supercharger can handle hot air just fine, it will be fairly dense hot air, but that is no problem. Just alter the drive ratio to get the required pressure increase. A few extra supercharger rpm is no big deal. Place all the charge cooling last, and pull out as much heat as possible there.
As for clutching and de-clutching, that too is something of very doubtful value. The manufacturers do it for two reasons. The Toyota blowers are rubbish, and wear out extremely fast if continuously driven. They also run very hot if driven continuously. Toyota clutch their blowers simply to make them last.
I am told Mercedes clutch their blowers for noise vibration and harshness reasons only. The clutches are a big problem, they operate under high stress, and figuring out an acceptable clutch control strategy to do this is far from simple.
Just use a solid well designed supercharger (most are) and keep it spinning. It simplifies things greatly, and there is no disadvantage except perhaps some slight continuous mechanical noise from the drive, especially if a toothed belt is used. Use a silent multirib belt if at all possible.