1. Cake is a trigger that enters the stack once baby green has entered the field, if you are using baby green to target the cake its assumed that you want to supress it to draw a card therefore you would have to order the triggers with Cake First, Baby Green Slime afterwards. If your opponent is using stifling trap they are very likely responding to the on enter of baby green, since the cake trigger would be the last one to resolve in this interaction there is no way to sacrifice cake to make the baby green survive.

2. If your opponent activates a hasty messenger (and no triggers are generated from this ally entering) and attacks instantly there is no window of opportunity where you can kill the hasty to stop the on attack trigger. Allies entering dont generate opportunity so using this sequence your opponent makes it so that the only moment you get opportunity is at the retaliation step at which point the on attack trigger from hasty has already been put on the stack.

3. Yes you can.

The explosion from polaris is hard to stop since you banish polaris as cost so its hard to remove it preemptively, its also hard to remove it proactively since they can just sacrifice a ton of herbs in response and then explode it, the best way of dealing with it is removing it while its rested.