Bear in mind that oil in this context is generally slow-burning lamp oil, not any kind of modern refined petroleum product. From a game balance perspective, a flask available for one silver piece shouldn’t compete in power with magic spells that do fire damage. So with that in mind:
Question 1: Lighting a square containing oil could reasonably light adjacent squares. Since a pint of oil will only cover a 5′ square to a depth of less than a hundredth of an inch, generally a single oil flask won’t really saturate a whole square. To have nine squares sufficiently oiled that they would all ignite as a unit, as a DM I’d expect many more than nine flasks of oil to be used.
Also, while I’d probably let all nine squares ignite at the same time, I wouldn’t want to extend this to arbitrarily large areas — at some point I’d give the “ignition wave” a movement speed (which also adds drama to the encounter).
Question 2: Correct, a creature standing in a square where oil is ignited doesn’t take damage immediately, and if on their turn they move out of the square (and into one without burning oil), they take no damage at all.
Question 3: Correct, a creature which moves out of the center ignited square and across another ignited square will take 5 fire damage.
Question 4: I would limit the damage to 5 points per turn. Consider that if a creature is doused in oil and then ignited, they take only 5 points of fire damage, even though they are (presumably) burning for an entire turn. Likewise, if they were to just stand immobile in a single ignited square, they would also take only 5 fire damage. My reading of the rules for fire damage from oil overall is that the intent is that being exposed to burning oil during your turn will do 5 fire damage. It doesn’t matter if the oil is on you, or if you run through one, three, or ten squares that are aflame: burning oil is hot enough to do some damage, but not so hot as to accumulate heavily in the space of a single turn.