The weird thing about "The Squire of Gothos" is that there are some very modern items there, such as the Salt Creature from "The Man Trap". If Jaeger was referring to the gothic architecture rather than the more specific items then the 12th-16th century time frame of gothic architecture (more like the 12th/13th to stay within the 22nd/23rd TOS century) would be reasonable.
As to the edge of the galaxy, "Miri" seems to suggest that Earth colonies and vessels didn't venture out more than "hundreds of light years from Earth". If that were the case, there might only have been a handful of unofficial attempts and no official attempts at leaving the galaxy by an Earth ship in the last two centuries for either technical, political or financial reasons. Other alien races might have tried (and failed) but they might not have shared those failures publicly.