It could be that your discs just suck, and you need a different brand. I haven't tried Verbatim CD-Rs on a PS1, so I don't know how they'd work for you; however, the pot on my dvd drive in my modified PS2 hated them. Ironically, the Sony CD-Rs seem to work fine for burnt backups, so give them a shot.
Just use Alcohol 120.. Burn the disc at 4x or 8x, use RAW DAO; keep in mind that some games use sub-channels, so if you're having choppy cut-scenes/audio, check "Rectify Sub-Channel Data." Also, instead of asking on a forum, why not just use Google..? If you're trying to burn a game that's bin/cue, select the .cue file in Alcohol 120. By the way, leave "Enable Buffer Underrun Technology" checked instead of disabling it just because you didn't see it written above. By the way, Alcohol 120 has a "Play Station" option for burning backups, but it pretty much just sets it to RAW DAO.
If you're still having issues after this, stop using discs that were meant for storing freaking pictures and buy a brand like Verbatim or Sony for your backups, sheesh.
And alcohol 120% won't find the .bin and the .cue so I'm stumped as to what to do
..Are you freaking serious? What do you mean Alcohol 120 won't find them, lol? Just go to "Image Burning Wizard" and manually select the .cue file, stop being lazy; the above portion of text applies to you as well. You're not going to be able to fit multiple titles on a CD, so that was silly of you to ask in the first place. Instead of using CD-RW, just get regular CD-Rs; I wouldn't be surprised if your PS1 refused to read the games from the CD-RWs.
Note:
If your PS1/PS2 isn't modified, then why the hell are you trying to boot backups in the first place? Also, your "techy" friend obviously isn't that technologically oriented if they tried for
hours and still failed to figure this out.. If you're still having issues after reading this post, maybe you should just give up and never touch the console again (If it comes to that scenario, sell the console to someone who knows what they're doing.)