No on both counts, pryman. The PSOne is only equipped with a CD-ROM drive. There's no way to make it play a DVD.
As for PSOne game downloads, the only ones that are legal are purchases directly from Sony's PlayStation Store (either on the PS3 or the PC- based version) for play on the PlayStation 3 or the PlayStation Portable. Anything else is completely illegal. There is no way to get a downloaded PSOne game to play on a PSone without modifying it. Best start looking around eBay for some of those older titles you want.