What exactly are the requirements to cast a 2nd level spell (which is what is being discussed, not 3rd) as a Sorcerer?
Depends on who you ask.
You defined "the ability to cast x level spells" as the ability to
produce an end result, like a Rogue using a Scroll can cast an 8th level Spell (like a sorcerer using a feat & item is really any different). Me and as the rules indicate pretty much everywhere define that term as the
granted allowance from a Class. You also, like most people that failed in math/sciences/computers, have no understanding of the difference between zero and nothing. For example if you look at the 13th level of Wizard they don't have zero 8th level Spells, they have nothing. A Fighter has nothing in regards to Spell access and a Versatile Grants nothing in regards to Spell access either. You assume they have 0 level Slots and thus can increase them using Feats and that is a logical error.
You really don't have to go any further than Practical Spellcaster. Everyone's favorite Feat because they cannot read past the first sentence, it's a prime highlight of exactly what I'm talking about.
Choose one 2nd-level spell from a school of magic you have access to. You gain an extra 2nd-level spell slot that must be used initially to cast only the chosen spell. Until your level is high enough to allow you to cast 2nd-level spells, you must succeed on a DC 8 caster level check to successfully cast this spell; if you fail, the spell is miscast to no effect. Your caster level with the chosen spell is your normal caster level, even if this level is insufficient to cast the spell under normal circumstances. When you become able to cast 2nd-level spells, you lose the benefit described above but retain the extra 2nd-level spell slot, which you can use to prepare or spontaneously cast a spell of 2nd level or lower as you normally would. Finally, you gain a +2 bonus on all Spellcraft checks.
If you're you then all you can see is the blue text and assume that you have "the ability to cast 2nd level spells" because, as you incorrectly understand it, you think it means producing an end result. But as the red text indicates, you do
not have "the ability to cast 2nd level spells" because the Feat does not grant any actual 2nd level Spell access in the way D&D uses the terminology.
And it's this way across the board, I didn't move a single goal post. Mage-Slayer does not remove that 15th level Wizard's ability to cast 8th level Spells anymore than a Barbarian standing next to them spamming Attacks of Opportunity does. The Wizard obtained his ability, sic access, and preventing certain uses of it doesn't remove this ability. It's the same with Class Features and Feats, heck even you use the term "inactive" when discussing Power Attack. You can acknowledge the separation between
having and
using it there, but not when it comes to Spellcasting because you'd be forced to admit you are wrong. So instead you're pulling a JaronK and trying to assemble components together to claim that you can meet the requirements that you've created as proof that you're doing things correctly and then you are using that to argue with.