Biochar is really just charcoal, made by heating without any/much oxygen in a process called pyrolysis. The benefits were first realized when ancient Indigenous sites were found in the Amazon basin containing stockpiles of it, and the Spanish found these sites to be amazingly fertile and productive. The normal soil in the Amazon is anything but fertile, as about 99% of the nutrients are sitting above your head in live growth. Nutrients get recycled fast in the rain forest.
Biochar really doesn't add any nutrients itself, but they are carbon sponges with each piece containing millions of microscopic holes, able to hold onto (bind) nutrients. The same property and large surface area makes them serve as a refuge for bacteria and mycorrhizae. Biochar is also considered a longer term sequestration of CO2, since it can literally last hundreds and perhaps thousands of years without oxidizing. So people are even looking at the possibility of using it to fight climate change.
No idea what the company "charged" their biochar with exactly, and if they don't disclose it, I would probably pass considering there are many sources of bacteria that will move into your biochar once your soil is established for any length of time.
The main problem with perlite is that it doesn't last compared to pumice and crushed lava rock. That being said, perlite is fine if you don't intend to be handing down your soil to your grandchildren, particularly if you're doing no-till. If you use a light mix with mostly peat like Pro-mix HP or something, yeah the perlite within the top inch of the soil will rise up if you flood the pots during watering. But it's only the top surface of the soil - don't worry about it. Preferably you should be using a mulch and/or cover crop over that anyway.
