The Bose unit in the spare tire area is a subwoofer. The QX30 came from the factory with run-flat tires. It sounds like you got a used QX30, so you'll want to double check the model of the tires on your vehicle to confirm.
The idle start/stop in my QX30 works really well. It's considerably smoother than the system in my Acura. However, everyone has a different "feel" for things like this. Without some more description, I doubt anyone can offer you an opinion on YOUR system. I suggest going to your Infiniti dealer and driving a NEW QX30 and see if it feels similar. If it feels the same, then you know that's how it works. If it feels different, let the dealer fix yours.
I would never buy a vehicle that only had one key fob. You really should have insisted that the dealer provide the second fob. The dealer can program new ones, but they're not cheap. If you go the eBay route, a locksmith would most likely be cheaper for programming, compared to the dealer. I still would expect you to pay $70-150 to an independent locksmith, plus the cost of the FOB.
2018 Infiniti QX30 Premium AWD - Majestic White
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