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Tengu
PFS Note The fan you gain from this feat is a held item (Pathfinder Core Rulebook 572)
You've learned to bind some of your feathers together into a fan to focus your ancestral magic. You gain a tengu feather fan: a magic item of light Bulk with a level equal to your level and the evocation and primal traits. You (and only you) can use the tengu feather fan to cast tengu magic. If your fan is lost or destroyed, you can create a replacement during your daily preparations; if you do so, your previous fan falls apart into mundane feathers. Further feats might grant you additional activations of your tengu feather fan, but you can never have more than three activations per day, no matter how many such feats you have. The DC for spells you cast with your tengu feather fan is your class DC or spell DC, whichever is higher.
You can cast the 1st-level gust of wind spell by activating your tengu feather fan.
Activate Interact; Frequency once per day; Effect You cast one of the spells contained in your tengu feather fan. Activating the fan takes the spell's normal number of actions. You can also Activate the fan to cast a cantrip you've gained from a heritage or ancestry feat; this activation doesn't count against the fan's uses per day
RAW, you can use your your Dex-based class DC (assuming it is better than your Cha-based 'Innate' Spell DC), but Charisma would still be your actual casting stat because there is no clearly stated exception to the innate spellcasting rules:
Core Rulebook pg. 302 3.0 Certain spells are natural to your character, typically coming from your ancestry or a magic item rather than your class. You can cast your innate spells even if you aren't a member of a spellcasting class. The ability that gives you an innate spell tells you how often you can cast it—usually once per day—and its magical tradition. Innate spells are refreshed during your daily preparations. Innate cantrips are cast at will and automatically heightened as normal for cantrips (see Cantrips) unless otherwise specified.Source
You're always trained in spell attack rolls and spell DCs for your innate spells, even if you aren't otherwise trained in spell attack rolls or spell DCs. If your proficiency in spell attack rolls or spell DCs is expert or better, apply that proficiency to your innate spells, too. You use your Charisma modifier as your spellcasting ability modifier for innate spells unless otherwise specified.
If you have an innate spell, you can cast it, even if it's not of a spell level you can normally cast. This is especially common for monsters, which might be able to cast innate spells far beyond what a character of the same level could use. Since this magic is innate, you can replace any material component with a somatic component (see Spell Components).
You can't use your spell slots to cast your innate spells, but you might have an innate spell and also be able to prepare or cast the same spell through your class. You also can't heighten innate spells, but some abilities that grant innate spells might give you the spell at a higher level than its base level or change the level at which you cast the spell.
If the feat said 'you can use your class's Key Ability Score as your casting stat instead of Charisma' then you would use Dexterity for your damage calculations.
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Class DC depends on the class. For example, Fighters, Monks, and Champions can use Dex or Str, chosen when you make the character, Investigators and Inventors use Int, Rogues can pick any stat other than Con, and Kineticists are the only class that uses Con.
Spell DCs typically only use Int/Wis/Cha, and a class typically only has one DC, unless they also get focus spells like Champions, Monks, and Rangers.
The spell damage however is still going to go off your Spellcasting stat, as the only thing that is changed by the feat is the spell's DC. The feat doesn't specify the fan's spells are Innate, so it's in a weird state, but it's best to assume they are. The fan however doesn't have cantrips in it, so the fact they aren't innate doesn't really matter.
It does allow you to cast any Innate cantrips you already have using the fan, but like I said in my first post, the only thing it changes is the spell's DC. It would have no effect on Produce Flame, and only make Electric Arc harder to resist. It wouldn't increase the damage of those spells.