If your nut is cut properly, and the neck installed correctly, neither of the problems you are concerned about should crop up. Locking tuners and a good nut slot should obviate the need for string trees. Assuming your neck is seated correctly in the pocket and the nut slots are correctly located, your strings should also stay on the neck, unless you are one of those who bends the E string away from the centerline of the neck, instead of toward it - in which case, consider a change.