Yes to both. If you're on defense generally you stay stickless and cover a guy at the point (blueline, furthest from the goal). If you're a defensemen and lose your stick a forward generally gives you theirs because you're more important and they play the stickless role covering lesser threats further away.

If you're not on defense, you skate to the bench and grab a new stick or jump off and another player with a stick jumps on.

Or, same player linked in article, Evgeny Kuznetsov, finds an opponent's stick caught in glass and takes it and plays with it (also happens to be a penalty, but a hilarious one) https://www.youtube.com/watch?v=jgYUdIDyNXg