Dauntless Bodyguard’s first ability isn’t a triggered ability and doesn’t use the stack. Players can’t respond to your choice of which creature it’s protecting.

If Dauntless Bodyguard enters the battlefield at the same time as another creature, that creature can’t be chosen for its ability.

If the chosen creature leaves the battlefield, you can’t choose a new creature for Dauntless Bodyguard to protect. If you activate its last ability in this case, no creature gains indestructible.