According to Memory Alpha, the NX Project was an attempt at breaking Warp 2.
...While in actual onscreen fact, the project was about breaking Warp 5.
The engine design was put through its paces like any engine in the real world, meaning it had to do Warp 1 at one point, and then Warp 2 at another, this latter point being shown in "First Flight" as having taken place in 2143. We don't know anything about the state of the art of warp engines in that year in general, though - many might have been doing warp three easily enough, and some Rigelian rentals might have moved much faster than Warp 5.
Should the
Franklin predate Archer's ship? Not until we hear more about this on screen. Could she? Sure. But if we take that one-sentence "backstory" from this unofficial source at face value, we still have to decide whether the class featured more ships than this one, and whether the
Franklin herself was the record-breaker, or whether the class in its entirety was designed for this speed.
As for USS or NX or the like, the one thing we know about those is that there is no rule, rhyme or reason to this in the 22nd century yet. Some ships get their registries painted on their hulls (NX-01, NX-02), others do not (
Intrepid. No prefix letters are painted ahead of the name on the hull, nor are they visible on shoulder patches. But computer readouts sometimes do show such lettering (
USS Enterprise and
USS Columbia are specified in "Divergence").
So, nothing exactly wrong with a ship launched in, say, 2148, and named
USS Franklin on the hull and the badges. That there would be personnel transporters aboard is in slight contradiction with "Broken Bow" where those are an absolute novelty to our space-savvy heroes. That there would be motorbikes aboard is only problematic in terms of user interfaces: Archer and Kirk both were unable to operate 20th/21st century automobiles efficiently. But perhaps motorcycle interfaces changed less, and nuKirk at least has the basic balancing act covered as he rides 23rd century bikes.
Timo Saloniemi