I can't seem to find a lot of information on this at all. While Admiral Yi's exploits are well-known, I can't seem to find a lot of information on the commissioning and de-commissioning of Turtle Ships.
I found one source that has the Kobukseon go all the way back to pre-Sejong days (1415) with early designs based off the Panokseon. Is that farfetched?
And what exactly happened to the Kobukseon after the wars? Joseon seemed to have an awesome naval advantage that (seems to me) could have been put to easy use. Or at the very least, continued to develop (would have helped Korea out when the West came knocking...).
Alex Hawley from the "Topics in Korean History Pocast" talks about the idea that Kobukseon has iron plating and spikes and points out all of the reasons why that wouldn't have been a reality. Including that the rumor was possibly originally thought up by Americans in 1866 hearing about the Kobukseon and immediately drawing connections to the recent Monitor battle that saw use in their own Civil War. This of course necessitates that there were no Kobukseon for them to see.
What gives?
In his book on the Imjin War, Hawley cites a 1413 historical record as the first textual appearance of the kobukson:
"While passing by Imjin Island, the king viewed a kobukson and a Japanese ship fighting against each other."
He describes the ship as having been developed to fight the wako pirates and then forgotten in the years afterwards. I'm afraid he doesn't provide any information that addresses your other questions, however; his last mention of the ships is during his coverage of Yi's death and his postwar discussion isn't particularly detailed.
I'm a little skeptical that the ships would have provided Korea with much of a naval advantage later, especially against the West. The impression I took away from Hawley's book was that while Yi was certainly a skilled admiral, much of the Korean naval success was due to Japanese ships being essentially little more than transports with little in the way of anti-ship weaponry. I'm not sure how well the kobukson would fare against an opponent actually using cannons.