How do I translate the words “player” and “team” into Mandarin?
Your thinking is correct!
班 is always 班级 (bānjí) — usually it means "class"

团队 (tuánduì) — you might consider the word as "group"

组别 (zǔbié) — also means "group", but it doesn't always have many people in the group

球队 (qiúduì) — sports team involving a ball, it's only used in ball games, like 足球队 (football team), 篮球队 (basketball team)…