It was surprisingly difficult to find solid data on this - individual play data is only readily available back to 1994, so consider this accurate up to that point. Anyway, the longest 3rd down conversion I could find was Viking Leroy Hoard running 53 yards up the middle on 3rd and 37 against the Broncos on 10/31/99.
The longest 4th down conversion since 1994 was the Bill's Rob Johnson throwing a bomb for a 40 yard touchdown to Peerless Price against the Colts on 4th and 34 on 9/23/01.
The longest down since 1994 was 3rd and 50 yards, faced by two teams: the Vikings vs. the Packers (12/20/99) and the Redskins vs. the Bengals (9/23/12). Neither was converted.
The longest 4th down since 1994 was a 4th and 48 by the Raiders against the Chiefs on 10/13/13. They unsurprisingly punted.
I can't find a way to filter for 3rd/4th and goal, but hopefully this at least partially scratches your itch. All plays found with this tool.