03 GT Heavy power loss when warm

Hey, i recently got my hands on a 03 GT for cheap, and of course as any cheap great car it was a can of worms.

At first car wouldn't warm up, but it was simply lacking a bit of coolant, no biggie.
Of course that was very likely intentional to hide the real problem : as soon as the thermostat open up, the car has trouble idling (and actually stalled on me once in a parking lot, it's an auto) and idle very low (400rpm) and any attempt to wot will only end up in sputtering.
It run great cold then as soon as it warm up it's crawling around, if i run it for 5mn and let it sit in a parking lot the next start will be rough.

Ran OBD2 check with a basic amazon tool, no codes at all. I wanted to check fuel pressure but late 03 have no port so i ordered a billet with a fuel pressure port. I have no real garage so i intend to change fuel and pump at the same time if fuel pressure is bad and only filter if it's good.

I have a little list of things that i should check, but most of this is pretty new to me as this is the first car i really work on (i have worked on carburetor bikes before but that's a totally different beast) so if anybody know a good & easy way to check for potential culprits (like a magic obd2 device that would give useful informations), i would be very grateful for it.
Any input appreciated, i will try to update for anything that i check, thanks !

First thanks a lot for the replies !
I ordered a bluedriver and it is indeed way better than my previous one the power off scan didn't find anything but the mode 6 did, rich to lean sensor threshold voltage(constant) failed for both cid 11 and 21. Suggested advice is to change O2 sensor and/or cat.
It also threw a fail on TID 2A and TID 2B wich may indicate misfiring but i'm not sure since i haven't found a reliable source on this.

I tried to log things but nothing go specifically out of spec, except TPS that doesn't go above 40% while i'm sure i opened wider, i also checked what i could but i see no damage to the wires, i didn't jack up the car yet so O2 is gonna be the next to check.
So indeed my next moves are gonna be a new thermostat, find a new jack, check O2 sensors, TPS, and probably coils & plugs.

Life happened so i did not do too much since then, except installing a nitrous oxyde rail adapter so i could test my fuel pressure, everything seem to be within specs so i'm crossing fingers, still gonna replace the fuel filter of course since it's quick & easy.
Jacklift should come on saturday so i'm gonna mess with 02 sensors. No further problems on the thermostat or the coolant has arisen, everything work normal levels are ok, but i'm adding the compression test box to my checklist, thanks !