I think that number was from the novelization of First Contact about the Federation fleet that intercepted the second Borg cube heading for Earth. The fleet managed to keep at the Cube all the way from the Typhon sector to Earth with Word and the USS Defiant engaged until the arrival of the USS Enterprise-E.
That battle was just months before the start of the Dominion War
Yes, circa November 2373 by most stardate estimates.
It's a long time since I read the novelization, but does seem credible. Only dozens of ships are seen at Earth itself onscreen, though.
As for USS Springfield NCC-1963, a Miranda or Soyuz-class seems highly likely. Kongo came from one of the list of unused Constitution-class names. And all ships on the Operation Retrieve chart have a Constitution-class logo (sans Excelsior, that is) - though as unused footage.
Helin makes sense by name as an Oberth (an astronomer) but only slightly less so by registry.
Plus TOS was in a period where numbers just might get reused too, alongside names...
