To answer the first question one has to look at the term Electrostatics.
Let's say that the field is not perpendicular to the surface. That means it has a component along the surface $->$ so now we have a force on the charges in the conductor along the surface of the conductor (no longer electrostatics).
Nothing is balancing the force, so it will then move the charge around. However, that means that our system isn't in equilibrium, since the charge is moving around. In equilibrium, the charges must be at rest, and that can only be the case when there is no electric force along the surface, i.e., when it's perpendicular to it.
Another, looser reason is, the gradient of a scalar field (eg. electrostatic potential) points in the direction of that field's greatest change. No change occurs in the field going along the surface, the gradient should not have a component in that direction.
To answer the second question one has to look at the intuitive understanding of field lines:
The electric field [and force] at a point is tangent to the field line that passes through that point. Does it make sense to you or is further explanation needed?