The widely used Kármán line definition used by Fédération Aéronautique Internationale (FAI) and the US Federal Aviation Administration is 100 km or 62 statute miles (54 nautical miles); only three X-15 flights exceeded that altitude, all before 1964 but two occurring after the end of the Mercury program The 80 km limit was used to justify giving “astronaut wings” to other X-15 pilots but is not universally accepted. @pkbites is correct that Melvill is the last solo American citizen in space at an apogee of just over 102 km.
125 km circular altitude is the lowest altitude where an orbit could be maintained for any duration, and the only solo flights to reach that altitude were Vostok and Project Vostok flights (not counting solo Lunar orbits during the Apollo Lunar missions, of course). Vostok 6, the first flight of a woman crew (cosmonaut Valentina Tereshkova) was in 16-19 June 1963, after the last Mercury flight (callsign Faith 7, 15-16 May 1963), so Vostok 6 would be the last solo orbital flight, but the o.p. is correct that Cooper was the last solo American astronaut in orbital space.
 I forgot about the Chinese space program. You are correct that Shenzhou 5 was the last solo orbital mission. 14 orbits isn’t much by modern standards but it is actually more than all but the last Mercury missions combined.