There isn't an explicit rule for this -- it is left to the DM to improvise an amount of damage that seems appropriate for the situation.

The DMG (p. 249) gives some guidelines on improvising damage. It suggests that falling into a fire pit might cause 2d10 damage.

In the case of someone running through a small area of fire as per your scenario, I'd look to the create bonfire cantrip from the Elemental Evil supplement:

You create a bonfire on ground that you can see within range. Until the spells ends, the bonfire fills a 5-foot cube. Any creature in the bonfire’s space when you cast the spell must succeed on a Dexterity saving throw or take 1d8 fire damage. A creature must also make the saving throw when it enters the bonfire’s space for the first time on a turn or ends its turn there.

Even though that is magical and not mundane fire, this suggests that a dextrous character could pass through a 5-foot cube of fire without taking damage, or with fairly minimal (1d8) damage if they're clumsy. Think about passing your finger through a candle flame without getting burned. I think I'd probably improvise pretty close to this for small areas of mundane flame. For a larger area, I'd probably do something like 1d10 per round in the flame.

The rules would have it do so, yes

The sorcerer’s elemental bloodline literally does nothing but swap the damage type of the spell; all other effects remain the same. In effect, an acid burning hands still launches a jet of flame and still sets things on fire – it’s just that the fire and flame now deal acid damage instead of fire damage. This can, of course, be easily justified as, “hey, it’s magic.”

Of course, seeing as acid’s effect is frequently described as a burning sort of feeling, this isn’t much of a stretch – the action to “put out the fire” could be “wiping the acid off” and “dousing” it could be “diluting” it. In most cases, “being on fire” is entirely equivalent to just “taking damage-over-time that you can use an action to end.” But then if you consider a spell that magically puts out fires – which would work – it gets a little tougher to explain.

It doesn’t have to be this way, of course. The DM certainly has purview to change something like this, and there is certainly room for adding variety this way. The problem is that the game does not define obvious substitutions to use; fire damage sets things on fire, but it’s less clear what the rest should do if you want them to all be distinct. Even setting up distinct damage-over-time effects is awkward – some spells douse fires, but there won’t be any spells that end these new damage-over-time effects that you make up.

Which is, of course, why the rules don’t do this: having a unique status effect tied to each energy damage type requires having the rest of the game react to that fact. Pathfinder doesn’t, and adding it now would be very difficult.¹ It’s much easier to say “it’s acid-damage-dealing fire, deal with it; magic, yo,” than to write out specific versions for each element, and then make sure they’re all reasonably competitive with one another, and that the rest of the product line takes into account, and so on.

1. ^{Remember, even when the elemental bloodline was written, much of “Pathfinder” was already “written” – the core rules did not change that much from 3.5, and most of 3.5 was just used as-is. A change like this would have required rewriting substantial portions of 3.5 that they chose to leave alone, and thus even at the “beginning” would have been very difficult.}