While it might not be ideal, my ranger's in the past have always walked around with bow in hand. Same way a wizard or monk with a quarterstaff would walk around with it out. Always bring those things up with the DM before game.
Training for years with a gun, bow, sword, shield, etc... it becomes natural to have them in your hand and on you. They become a part of you. An extension of who you are.
Going even further, all rogues I have ever played walk around quietly (taking penalties where appropriate). Not hiding, but not making much noise either. Training for years to move quietly, it would become natural to always walk that way. Its called muscle memory. When a movement is repeated over time, a long-term muscle memory is created for that task allowing it to be performed without conscious effort. This applies to controlling a horse with your knees.
I don't see a problem with being mounted and having on hand on your bow or crossbow across your lap, and one hand on the reigns. In fact I would assume that all of the party exploring a hostile territory would have their weapons ready at all times.
As far as the NPC thing goes, I personally cant see a NPC getting their panties in a bind because someone is walking around with a bow or quarterstaff out. Now a bunch of adventures with swords drawn out over their shoulders and at their sides might cause NPCs to get a little jumpy.
On a side note, I have been bow hunting. Carried that thing with me in my hand all day long.