In the opening tipoff to start the game, can they only stand on the opposite side of the halfcourt line? And for the other tipoffs I feel like the players on each team always stand in the same place (right?) -- are there rules that force them to do that? Or could they stand wherever they want?
FIBA rules state (Article 12.2.2):
Team-mates may not occupy adjacent positions around the circle if an opponent wishes to occupy one of those positions.
NBA rules say essential the same thing, although in a little more detail (Rule 6, VII):
g. The eight non-jumpers will remain outside the restraining circle until the ball has been tapped. Teammates may not occupy adjacent positions around the restraining circle if an opponent desires one of the positions. No player may position himself immediately behind an opponent on the restraining circle.
h. Player position on the restraining circle is determined by the direction of a player’s basket. The player whose basket is nearest shall have first choice of position, with positions being alternated thereafter.