It matters hugely. It's part of my core argument. If you have multiple sources you can be opportunistic. If you get a boost from 30% to 50% take it and move on.
Exactly the same regardless of chemistry. Obviously if you have a source of charge, use it.
I have as much solar as will fit on the available deck space. It contributes a modest amount, but nowhere near the daily consumption. In Alaska last year, with rain nearly every day, not that much at all. The alternator regulator will communicate to the BMS, but that is a very small addition to function in my view. Primarily it serves to kill the alternator output when the BMS disconnects, saving the alternator diodes and preventing the voltage spike. The same function is served by a FET isolator to an LA bank, very likely more reliably. And the latter (but not the former) also solves the backup power problem.You have a genset but really you'd rather not have it. And you haven't acknowledged solar in the mix. For sure you should charge with that $800 setup. Make sure the BMS communicates.