Sometimes, it depends on the modchip, too. Because of security protocols, depending on how you power on your console, you enable or disable the modchip.
Most modchips just have you simply press power for standard use with the modchip in enable mode. Pressing eject to power your console from standby puts the modchip in disable mode (for playing retail games with online capabilities). Powering on the console with the power button held for 5 seconds allows for "stealth" PSOne mode. This is how it was with my PS2 modchip. If you know what's installed in the PS2, you may have to Google how to do enable/disable procedures for it.
Verbatim discs, usually the gold standard for game backups, are great. But I've used Sony, Memorex, and TDK discs with no hiccups at all. DVD-R's, again, are standard, but I've used DVD+R's with absolutely no problems whatsoever. The last backup I made was over 6 years ago with a +R disc, and still today it plays with no problem.