Rules as written, yes, numerous tricks are available here.
Precocious Apprentice gives you a real 2nd-level spell slot. It has a special restriction that it can only be used for the chosen spell and requires a caster level check, but it is otherwise a spell slot with all that entails.
Focused specialist replaces a spell slot. As established, the spell slot from Precocious Apprentice is a spell slot—it has an additional rule, but is lacking nothing of “spell slot-ness.” Moreover, the replacement effect has says nothing about either not working on spell slots with special rules, or about transferring those rules to the two new slots you gain.
As a result, as written, the result is that you replace your singular 2nd-level spell slot, that would only be available for the spell chosen with Precocious Apprentice and require a caster level check, for two spell slots, which can be used for any spells of your chosen school, without a caster level check. Metamagic’d spells are still of your school, so yes, you could apply Sculpt Spell to something to make use of those spell slots.
Of course, though, you don’t have to. Wizards are welcome to scribe spells into their spellbook even before they can cast them. And then once you get those spell slots (normally by leveling up, but in this case by via shenanigans), et voila! you can cast them.
In addition to the straight benefit of giving you twice as many 2nd-level slots and getting around the limitations of Precocious Apprentice, some argue that Precocious Apprentice is insufficient to meet requirements that include “ability to cast 2nd-level spells,” focusing on the plural on spells and the fact that Precocious Apprentice only gives you one slot, for one spell. This is, in my opinion, invalid, but if you had to deal with such a restriction, focused specialist would get around it.
On top of that, the feat Versatile Spellcaster can be taken to “combine” two spell slots of a given level for a single spell slot of the next-higher level. Since Precocious Apprentice and focused specialist result in two 2nd-level spell slots, you could then use Versatile Spellcaster to combine them into a 3rd-level spell slot. Like focused spellcaster, Versatile Spellcaster lacks any rules about transferring restrictions from the spell slots you use to power it, so your 3rd-level spell slot is free to use for any 3rd-level spell. And again, metamagic or just 3rd-level spells are available to use this spell slot.
(You would have a hard time—though it’s probably not impossible—to actually use Precocious Apprentice and Versatile Spellcaster on a 1st-level wizard, since Versatile Spellcaster requires spontaneous casting. There are ways for a wizard to get that, but without using more feats, which you don’t have barring flaws, or more levels, those aren’t available.)
These kinds of abuses are commonly used in theoretical optimization builds to meet prerequisites ahead of schedule—so-called “early entry” into prestige classes. Depending on how far you want to abuse it, you can literally get epic spells at 1st level (Dragonwrought kobold to take epic feats before 21st, worshiping elder evils for bonus feats, dark chaos feat shuffle to make better use of your feats, these and other tricks for increasing spell level, bags of rats with greater consumptive field for caster level, various other tricks for skill ranks—its all possible). But because such abuse is certainly possible (and we can always cut out a lot of busy work accomplishing this, and just go for Pun-pun), it all becomes rather pointless.
Instead, limits are placed on these kinds of loopholes and abuse to prevent these kinds of things. If all you want to do is get a couple of spell slots to use with Sculpt Spell, your DM may be OK with that—I probably would be. But I would recommend not allowing, or requiring, these kinds of tricks to do it. If it’s OK for the game, I would just make a houserule for how to do it—and then ban these kinds of tricks that can go so, so much farther.
