Have you looked at the WS2812(B) LED strips. They have controllers built into each RGB "pixel" so you can control them individually. You use one Arduino pin to control many many (100's) LEDS wired in a "daisy chain". The FastLED library makes programming sequences (with fade, blink, etc) pretty easy.
Backstory: my gf wanted RGB computer lighting. I looked at the costs and the only way I could see it done was for $200-240, which isn't justifiable.but it might be a fun project to do for about a $100 arduino project that's truly customizable.
(have written mostly c++ which I heard is what uno's use) so I feel I can manage this.
Additionally I do have access to an professional for help if necessary (my father is an electrical engineer by trade, I don't want to approach him for help with a half-fledged idea for something of non-practical purposes)
around 50 general purpose transistors (one for each colour of the RGB's)
Additionally I'd like to house everything in a drive bay (like below a dvd drive), from what I've seen all these components should be able to fit in something that size.
This seems feasible but I'd recommend that you start with 1 or 2 LED strips to get the hang of how everything connects together. Maybe you can get that done in 6-weeks. In fact, you should probably start with a couple of regular-little RGB LEDs so you don't have to worry about the addressing or drivers. You can do a lot of preliminary software development and work on the user interface without the LED strips.Do you need dimming? It's easier to "expand" to more channels if you don't need dimming.Or, there are addressable LED strips with drivers built-in. (Check the Adafruit website.)
I doubt you can do it for $100,
That's probably "pushing it" when you consider the connectors & driver circuits, etc. And, is the power supply supposed to fit in there? The smaller you try to make things, the trickier it gets, especially for a hobbyist.