Are you looking to record and stream live ? Or record, edit and upload as a finished video product ?
Your Fuji will likely work fine, but if not, then even a cheap option could be basic modern-ish camcorder and connecting an hdmi cable into an hdmi usb capture dongle on the computer. I found these can make for decent video quality. Camcorders can zoom easily to adjust their image position, they can easily be mounted on stands / arms, and they also have screens to flip towards you so you can look at the camera itself and immediately see the image. Great for when you are talking to camera pieces.
In terms of lighting, in my office I have a window directly behind me which makes me silhouette against it when on a webcam call. I bought an under counter strip light similar to something like this: ( found from a quick google search)
Natural White (4000K) LED, ideal for any style of Kitchen - 16W | 1600 Lum | 240V- IP20 Rated- Can be interlinked using accessories (sold separately)- Can be wired to mains wall switch or use Rocker Switch- LED Strip Light, 1.5M Mains Cable
www.toolstation.com
which i plonk somewhere by the side of the monitor to light up my face with a large diffuse light. It’s not adjustable Brightness, but moving it about / making it face the wall affects the light on my face well. it’s cheap as chips, doesn’t weigh much, and plugs straight into the mains and easily moved about. Makes a massive difference to my webcam stuff.
As for mics … sorry, not that clued up on them.