I never changed CD drivers on any Windows for any drive. They just work. If not, they're fucked.
NEC CDR-1901A CD-ROM is actually quite googleble - it's a CDR drive, so not really ancient. There are even IDE drives from 1994, those are ancient.
Basically, get another one that looks good in the machine, maybe for that era, some that says "48x" on the lid would be period correct and authentic. Should cost around one buck, the challenge is to find a working one.
Screen: Hell, is that a 4k TV?
Don't expect too much, that's just several quantum leaps apart.
Cheap converters like that, you're lucky if they handle one resolution correctly - probably it will be 1280 x 720 or 1920 x 1080p full HD.
People keep CRT screens just for the resolution issues.
Not sure about modern TVs, it may vary by make and model - do those still have ANY legacy input below HDMI?
Like, composite, scart, s-video, a bloody yellow RCA, if only via some finnicky breakout cable?
All of those may give you better results with a VGA card with good TV-OUT chip...
Else, sorry, once the signal is HDMI, it's HDMI, no way changing it on the TV. So, if the adapter doesn't allow you to set it right via some input, AFAIK, you'd need a different adapter.
Someone will have had the same problem, but in general: Translating an analog, fairly high res video signal like VGA to an digital format is NOT trivial, professional gear for that costs four figures. Consumer stuff like that, you can only try out. Maybe some driver tweaks will alter the signal to another frequency that will end up looking better, but it'll be a stab in the dark.
Are there adapters with, say, USB hookup and some kind of client software to customize them? Only solution I could think of that will get it right.
So many slightly older TVs used to have native VGA input...
Sorry about the voodoo card, but with a AGP rig like that one, you could get a geforce or radeon with DVI out - that's HDMI by another name and chances are it will look good on a TV screen. 😉