If it's true mahogany, no stain at all is needed. It would obscure the beautiful grain. Look at pictures of antique Chris Craft boats if you want to see some beautiful wood in it's natural state. If it's luan, it won't matter what you do to it.

I wanted to add that we will be Lime Washing the exterior brick so it won’t be pink

You are sure this is what you want? I don’t see it on your house. But let’s say you go with it. What would you do with the vinyl garage doors?

It is always a better choice to use solid wood for an exterior door...in your case there are no exterior wood features so painting the door would be the most practical solution for a veneer door...it would be tough to pick a decorative color for the door in this situation...so matching the white trim or picking a color from the stone colors would flow best...