It has been a while since I modded my PSP so forgive me if I get some info wrong.
To my knowledge PSP Eboots are the format for programs to run natively on the PSP. That is just how the PSP knows that a program or game is made to run on it.
To run the Eboot PS1 games from this site and any emulators on your psp you will need to soft-mod it with custom firmware.
There are plenty of tutorials on YouTube on how to do this.
Below is a link on how to install custom firmware on you PSP:
The custom firmware version I’m on is 6.60 PRO-C, but I modded my PSP before the switch was released so I’m sure there is a more up to date version or a new custom firmware entirely.
To reiterate, YouTube tutorials are your best friend for these types of things.
I recommend getting a Memory Stick to Micro-USB adapter so you can have as much space to store games, themes, etc. on your PSP. The PSP was made with proprietary Memory sticks the largest was up to 64GB of space.
Using a Memory stick to Micro-USB adapter allows for more space to store whatever you want. Just make sure that the Micro-USB you use doesn’t exceed 128GB or else the PSP will think the Memory Stick is corrupted or one is not inserted
Before you put anything on your PSP memory-stick you have the format it, this can be done after inserting it into your PSP and going to your setting and selecting the option to do so.
Link for Memorstick to Micro USB adapter:
https://www.amazon.com/Funturbo-MagicGate-Playstation-Portable-Handycam/dp/B07F68FCHD/ref=sr_1_1_sspa?crid=2QNGIS3VZ6O1I&dib=eyJ2IjoiMSJ9.ii9aU_sWyn9XJZpRhKCBqBSUvYge88ZF22pjnd3lPtkDW0YTNLkYbWJd0ba2rSP5Sd8wZslNbrTlw1NOJ1gOAlD1vqUF07XmXpekE_P1Fvzwr1U2qA4hWCK5I64R8v5qxsUBEYme2wyD6mZ3vkFhTxQfn9HWyfJDwH73wjx9uBC6oULv3eCtkBAE4OKFroBJlFYLBHLxfI-cGagkcEqawSJaVI9FnX_xd0VDwIRPkIg.v9ZRcepDQnxXERu2mlDKhhikkJ7-6SwSN9iKLkJ3iKY&dib_tag=se&keywords=psp+memory+stick+adapter&qid=1757893844&sprefix=psp+m,aps,139&sr=8-1-spons&sp_csd=d2lkZ2V0TmFtZT1zcF9hdGY&th=1
The link below is a site with custom PSP themes:
https://www.pspunk.com/psp-themes/
This second link is a tutorial on how to install custom themes:
https://www.pspunk.com/psp-cxmb/
For PSP and PS1 games you can stick them on your memory card after installing custom firmware and they will run and you can play them as long as the PS1 games are in the Eboot format.
For other consoles there are emulators made for the PSP to run them. There are a few for each console but I personally have:
TempGBA for GBA games, PiceDrive for Sega Genesis, Nester J for NES, Master Boy for GB and GBC games and SNES9xTYLmw Mod for SNES games.
Also to get to the repo and look at the section of games click “The Repo” button at the top of the page. From there go to the Platform tab in the top right corner, hover your mouse over Sony, then click either PSP or PSP Eboots to look at games for your PSP.
I don't know if there are emulators to play other console games on your PSP in the repository, I got those from other sites.
Hopefully this helps and answers all your questions and then some. If you need something else or have other questions I will try to answer them to the best of my ability.
EDIT: Also if any of this info is wrong, feel free to correct me.