Does the castle style I choose at the beginning of Warrior's Way have any impact on the gameplay/sidequests/etc of the game? Or is it just about aesthetics?
You have 3 choices of castle, but they are purely aesthetic choices - only upgrading your castle affects gameplay.
When you upgrade your castle, you gain perks at the cost of converting some of your warriors to craftsmen. The perks you gain are detailed on this answer.
To upgrade your castle, you must come across some 'resources' after defeating a nation (which look like a pile of wood and bricks). You will then get the option of upgrading your castle.
For completeness sake, the choices of castle you have are:
You also cannot alter the choice of castle after you've chosen it.