BT points are points you get from completing quests and are relevant to certain events, like this round's panel event. You will not see BT points accumulate on your quest results screen, but you can check where you're at on the event page's ranking list.
Yellow cards gained during training are cards that are rarer "pulls" during that training level. They're usually one star higher than the other three cards. No, they aren't required for boss fights or loot. Most people use them for feeders (cards used to enhance other cards).