Your first option is correct - you choose from the options listed in the "Key Ability" list of the class in question. Rogue is slightly different because its Key Ability is listed as "Dexterity or Other", where "Other" is clarified to mean "an option from rogue's racket", and since you do not get a racket, you have no "option from rogue's racket" to choose, requiring you to choose Dex.
There isn't any specific rule I can find that states this unequivocally, but per the quote you include concerning Class DCs, it is equal to 10 plus proficiency (2+level for Trained) plus the modifier for the class's key ability score, which is the exact words used in the class features list I have called out above, as well as clarified in the description for Key Ability, so I have a high level of confidence in this interpretation.
You have two separate spell slot lists. This gives you
Wizard Spells
+-------------+-----------------+
| Spell level | Number of slots |
+-------------+-----------------+
| 1 | 4 |
| 2 | 4 |
| 3 | 4 |
| 4 | 4 |
| 5 | 4 |
+-------------+-----------------+
and
Sorcerer Spells
+-------------+-----------------+
| Spell level | Number of slots |
+-------------+-----------------+
| 1 | 2 |
| 2 | 2 |
| 3 | 2 |
+-------------+-----------------+
This gives you a total of 6 first, second, and third level spells, spread across different sources (so the slots aren't interchangeable, but there are still 6 of them)!
The Wizard spell slots can only cast spells from your spell book, and your Sorcerer spell slots can only casts spells from your sorcerers repertoire.
Relevant rules:
Spellcasting Archetypes (emphasis mine):
... All spell slots you gain from spellcasting archetypes have restrictions depending on the archetype; for instance, the bard archetype grants you spell slots you can use only to cast occult spells from your bard repertoire, even if you are a sorcerer with occult spells in your sorcerer repertoire.
This means we have to look at the specific archetype for the restrictions, but the bard example gives us a pretty good idea of what to expect. So, looking at the Sorcerer dedication:
You cast spells like a sorcerer.
From Sorcerer's Spell Repertoire:
The collection of spells you can cast is called your spell repertoire.
So you can only cast spells from the archetype that are from your Sorcerer spell repertoire. It's a little round about reading to get unambiguous RAW (rules-as-written), but essentially multiclass archetype spell slots do not stack with other spell slots unless explicitly mentioned.
Best Answer
Yes, but maybe only with retraining
The rules for Retraining indicates that it is possible to choose to do so...
The Archetype gives you a spell repertoire, so it stands to reason that guidelines for retraining still apply, even if you don't have Sorcerer Spell Repertoire class feature itself.
I'd go a step further
I would follow all of the rules of the spell repertoire class feature for the archetypes that indicate "You gain a spell repertoire..."; all of it's abilities and limitations, except modified by the spellcasting archetype rules limiting spell slots and spells known differently than the original class.