Vedanta has a rich tradition of rational thought, debate etc. and they mock other philosophical systems that are merely belief systems without any rational thought or believe in amassing numbers to show their might or legitimacy.
As regards believing anything: on the contrary traditionally everything was questioned and settled through a debate (which had precise rules and thoroughness that modern scientists cannot even dream of). Unfortunately, the spell of abrahamic religions destroyed that to an extent since they themselves are merely belief systems without any culture of civilized debate.
As for Shri Ram as incarnation, it would be good for you to provide the source of your information that He incarnates to study the difficulties of mankind -- this would be extremely stupid indeed. The reason for His incarnations are to re-establish dharma, punish the wicked, deliver the pious devotees and so that the devotees can taste the nectar of His lilas (Bhagavad Gita 4.7-4.9).
