The fan does not bother me at all

Yes indeed, but it depends mainly on the television. If it is recent, it will have an HDMI output which will allow the sound to be transmitted and the bar to be controlled via the television (louder, softer). If not, it will be connected the old way.

Hi I have connected the soundbar to a TV via hdmi arc and the sound always comes through the soundbar, whether I watch TV or turn on the Xbox or the blueray player. So I assume it should be the same with the projector, provided all your external devices are connected directly to the projector. In principle, it should be the same and therefore work.

I also bought this product and it does indeed match the video. The dimensions of the sound bar are not the same as the sound bar (108mm length stated when it is about 90). For me, it played better for its location, so no inconvenience.

Cables are very long, I don't know exactly but 10m for sure. The only problem with this system is the cool fan in the woofer, which always makes a lot of noise.

No. According to Sony, the following scope of delivery is included: Optical cable, colour label for box, batteries, warranty card, quick configuration.