Policy dependent on initial state distribution in finite horizon MDPs
On an MDP the main assumption is that states are independent (Markov Decision Process). Any policy in such environments will decide an action on a given state independently of previous visited states, taken actions and received rewards.
This means that the optimal policy doesn't depend at all on the initial distribution. However the initial distribution might affect the expected reward but it won't affect the optimal policy.
To be clear on your second question, the condition that allow us to rule out this dependency is the markov property: The future is independent from the past given the current state. So the optimal policy will have a distribution of actions to take depending only in the state, without caring about the weight of such state in the initial distribution.