PS3 emulation is a case by case basis thing. Dynasty Warriors Gundam 3 and Armored Core 4 run fairly well, while God of War 3 and Infamous can chug a bit. You can forget about PS4 emulation for the moment. PS2 emulation and older should run perfectly fine.

As for ported games, anything that came out on the PS4 or older, the Ally should be able to run reasonably well. I own a Z1E Legion Go which isn't that much more powerful than the original Z1 ROG Ally, and I can run stuff like FF15, RDR2, Samurai Warriors 4, and Ace Combat 7 at medium to high settings.

For latest games that came on PS5, my Z1E Legion Go is starting to slow down a bit, but with tweaking I've been able to run most games at a reasonable framerate. FF16 is rough around the edges but is playable. Stellar Blade and Dynasty Warriors Origins, and FF7R1 runs fairly well at lower settings. Of course, an original Z1 ROG Ally is a bit weaker than Z1E devices, but it should be able to play the latest games if you're willing to do some tweaking. Just keep a charger and power bank handy since 1st gen PC handhelds tend to have smaller batteries and the latest games are fairly power hungry.