1) With the caveat that I don't use AF, the Panasonic S5 is a wildly good value at $800 used (what I paid for mine). Imo, the improvement in noise and dynamic range over the GH5 is worth it, but if you're always in well-lit areas maybe you won't notice much benefit. If you aren't tied to existing lenses, camcorders can be more convenient than hybrids. I'm not familiar with available models since I don't shoot events, but I'd research camcorders for event videography with no photos.
2) C400 or FX6. The C300 Mk III is also fine--I choose full frame because of the lenses I own. It also depends on how much weight matters, as the FX6 is extremely light. If I need photos as well, then add a photo body in addition, and honestly everything is so good these days that I'd take anything modern.
3) Assuming 1-3 person camera crew, it depends on the script. My general pick right now for myself is a C500 Mk II (cheaper than a C400). I used to have Z Cam, which has unbeatable wireless monitoring and control, but I don't like the lack of color management support in Resolve. I would pick a Z Cam F6 if I need to fit the camera into tight spaces. If it's a lot of fast action like a car chase, a Komodo's global shutter would be great. The Ronin 4D is also very interesting for extensive handheld--never used one myself, though. If I can step up in budget, I'd love to try the Blackmagic 12k LF.