I've been able to improve on Etznabihik Mangk but in a different category. In this category the record should go to oroboras for his SLULEE DN-G B13-3 A 7 at 4,071 km from
here.
SMALLEST TERRAFORMED BODY - PRIVA A 1 WITH A RADIUS OF 2,670 KM
LIGHTEST TERRAFORMED BODY - PRIVA A 1 WITH 0.0718 EARTH MASSES
LARGEST TERRAFORMED BODY - 13 ORIONIS 4 WITH A RADIUS OF 8,935 KM
HEAVIEST TERRAFORMED BODY - 13 ORIONIS 4 WITH 3.8340 EARTH MASSES
HOTTEST TERRAFORMED BODY - 13 ORIONIS 4 WITH A SURFACE TEMPERATURE OF 355 K (that's hot and it's not even a penal colony

)
TERRAFORMED BODY WITH THE HIGHEST SURFACE PRESSURE - FRAGLAND IN COCIJO WITH 6.32 ATMOSPHERES
There is no such category as below but, probably, there should be for Earth-likes and terraformed bodies (and for water and ammonia worlds probably, as well):
TERRAFORMED BODY WITH THE LOWEST SURFACE PRESSURE - NEW WORLD IN ACHENAR WITH 0.25 ATMOSPHERES
COLDEST HIGH METAL CONTENT PLANET - ALCHITA B 10 WITH A SURFACE TEMPERATURE OF 49 K
LARGEST ROCKY ICE PLANET - PSI TAURI 6 WITH A RADIUS OF 11,223 KM
HEAVIEST ROCKY ICE PLANET - PSI TAURI 6 WITH 7,2983 EARTH MASSES
HOTTEST ROCKY ICE PLANET - PSI TAURI 6 WITH A SURFACE TEMPERATURE OF 1820 K
ROCKY ICE PLANET WITH THE HIGHEST SURFACE PRESSURE - PSI TAURI 6 WITH 784,120.00 ATMOSPHERES
HEAVIEST GAS GIANT- ACHENAR 9 WITH 4,893.9912 EARTH MASSES (how do you find this one?)
HEAVIEST CLASS IV GAS GIANT - ACHENAR 9 WITH 4,893.9912 EARTH MASSES
The record of smallest gas giant from Hyperion is pre-Gamma so it should be removed (Hyperion is entirely different now) - I have found a nice replacement:
SMALLEST GAS GIANT - AETERNITAS 5 WITH A RADIUS OF 10,658 KM
LIGHTEST GAS GIANT - AETERNITAS 5 WITH 1.4403 EARTH MASSES
SMALLEST CLASS I GAS GIANT - AETERNITAS 5 WITH A RADIUS OF 10,658 KM
LIGHTEST CLASS I GAS GIANT - AETERNITAS 5 WITH 1.4403 EARTH MASSES
LIGHTEST CLASS III GAS GIANT - HIP 63422 10 WITH 4.9693 SOLAR MASSES - this record should be removed as there is an obvious bug (the planet cannot be both class II and class III - it seems more like class II), look at the picture
here. I have a replacement:
LIGHTEST CLASS III GAS GIANT - 63 CHI LEONIS BC 1 WITH 9.7772 SOLAR MASSES
SMALLEST CLASS III GAS GIANT - 63 CHI LEONIS BC 1 WITH A RADIUS OF 14,474 KM
SMALLEST GAS GIANT WITH WATER-BASED LIFE - GATO 9 WITH A RADIUS OF 15,206 KM
LARGEST WATER GIANT - EPSILON SERPENTIS 10 WITH A RADIUS OF 25,565 KM
HEAVIEST WATER GIANT - EPSILON SERPENTIS 10 WITH 218.9143 EARTH MASSES
SMALLEST WATER GIANT - HIP 110773 8 WITH A RADIUS OF 16,689 KM
COLDEST RINGED STELLAR BODY - NAREN 11 WITH A SURFACE TEMPERATURE OF 270 K
YOUNGEST RINGED STELLAR BODY - NAREN 11 WITH 868 MILLION YEARS
SMALLEST Y-TYPE STAR - HIP 110773 13 WITH A SOLAR RADIUS OF 0.0438
HOTTEST Y-TYPE STAR - RA 12 WITH A SURFACE TEMPERATURE OF 677 K (temperature not pressure as in the previous record)
OLDEST Y-TYPE STAR - MAHER IN I BOOTIS WITH 15,000 MILLION YEARS
OLDEST K- TYPE STAR - ERAVARENTH B WITH 17,132 MILLION YEARS
OLDEST G-TYPE STAR - I BOOTIS C WITH 15,000 MILLION YEARS
SMALLEST A-TYPE STAR - WO 9733 B WITH A SOLAR RADIUS OF 0.9718 (another one bugged in the nav search and I've forgotten to take a picture in the Galactic Map so I don't know what the star designation is - it certainly doesn't look like A-type)
LIGHTEST A-TYPE STAR - WO 9733 B WITH 0.9531 SOLAR MASSES
COLDEST A-TYPE STAR - WO 9733 B WITH A TEMPERATURE OF 5,680 K
COLDEST WHITE DWARF - LHS 6309 A WITH A SURFACE TEMPERATURE OF 6,106 K
I've confirmed the smallest star record and I also claim in the name of Rhysling (great find!):
SMALLEST WHITE DWARF - CHARICK DRIFT A WITH SOLAR RADIUS OF 0.0008
HEAVIEST WHITE DWARF - CHARICK DRIFT A WITH 1.4336 SOLAR MASSES
Edit:
HEAVIEST METAL RICH PLANET
S171 37 8 AT 1,397.9980 EARTH MASSES
HOTTEST METAL RICH PLANET
S171 37 8 WITH A SURFACE TEMPERATURE OF 33,501K
METAL RICH PLANET WITH THE HIGHEST SURFACE PRESSURE
S171 37 8 WITH A PRESSURE OF 38,894,530,560.00 ATMOSPHERE
Even if we assume that the values should stand (and I think they should to make them more visible by FDevs) you still have it wrong - this is a high metal content planet!
Edit 2:
HOTTEST T- TYPE
NU FORNACIS C WITH A SURFACE TEMPERATURE OF 10,334.00 K - this is also wrong as NU FORNACIS C isn't a T-type star but it's a T Tauri