You can use nearly any period (read: basic 8-32x CDROM only, no CDR or DVD) ATA CDROM drive in a 4400, but you'll most probably need something like CDROM Toolkit to get it to work once Mac OS loads; it should still boot from CD regardless. Try to go with brands such as Matsushita/Panasonic, Sony, Hitachi, Pioneer, Toshiba, or NEC for best compatibility; sometimes drives from less common brands such as Mitsumi, Lite-on, Philips, and cheapies like ACER drives don't work well in a Mac, and some brands like Yamaha and Ricoh have weird drive trays and buttons that don't line up with the Mac's bezel properly.
What brand of USB card are you using? Macs will only use OHCI-compliant cards. The safest bet is a USB 1.1 card with an OPTi FireLink chipset since this chip was used onboard several Macs. Anything with an NEC chip should work, too. Avoid anything from VIA because they don't work at all in a Mac (and often don't even work properly in a PC either). This also goes for FireWire cards: try to use Agilent/Lucent or TI chipsets and avoid anything from VIA because they aren't going to work.
Did you try posting in the Trading Post for a keyboard and mouse? Someone may be able to get you a set for a reasonable sum.
That's bad luck.
I am no expert but would myself try to confirm that the RAM is good first, as it is possible that you just have two sets of bad ram. After that I would suspect the analog board in general and the flyback transformer in particular, given the varied square and herringbone patterns.
There are others on here who are much more knowledgeable technically, however, who will hopefully chime in, though recurring questions get a little wearing to answer repeatedly....
Some resources that you may or may not know about in the meantime:
I installed the 8MB of RAM I bought for it and now it does this, after displaying vertical lines for a while. Reset sometimes makes it change between vertical lines and this. I've also installed a new PRAM battery.