1) What does Anne thing their helpers do to "prove" their heroism?

An inference that can be made is that the princess is bothered about the tiger door. although there are reasons for both this one has a better reason. the princess is not a monster and even if she didn't want to marry this man she wouldn't want him to die. the grudge of death is pretty heavy. unless she has done this before and feels no regret then she wouldn't want to be married to someone of low quality, if he is low quality.