Yes, there are lots of ps1 emulators for psp. Sony's official emulator is not so reliable. But choosing the best emulator is difficult task. Here is full detailed article that you should read to get best ps1 emulator for you psp.
The answer to the second question is that yes ps1 emulator offer video output but some emulators don't offer video output so you need to choose best for you.
Answer to 3rd question is while there's an option of remapping the controls however you want, generally, you'll assign either the R2/L2 buttons or R1/L1 buttons to a D-pad press or one of the directions on the analog stick (depending on which control scheme you've chosen). If you have a PSP Go synced with a DS3/Sixaxis, it will use all the buttons correctly, which is great if you're outputting to a TV.
To get answer of your 4rth question, you must see this video:
https://www.youtube.com/watch?v=kxMH67yvEqo
