Proxmox is based on Debian the way Ubuntu is, but more highly focused and without departing from main Debian strategies.

Proxmox is a kernel and set of tools for managing virtual machines and containers, in this case LXCs. These are different from docker, they share the kernel with the host but have isolation at namespaces (user id's, process trees, network views, etc). They boot in seconds because they bypass hardware virtualization like you'd have with a VM.

I'd say gaming is better off as its own project effort. You could do things like GPU pass through and a dedicated VM for gaming, but you would be adding a pretty hefty layer of complication. If you dont need to convert av1 to something else, I'd just go with a 7th+ gen iGPU, and dont worry about it. Mine are 7th-10th gen, I have dozens of TBs of media, and they just chug along without issue handling somewhere between 4 and 6 streams at typical top use, mostly streamed without transcoding anyway. AV1 for the source file, you'll be transcoding almost everything because most devices dont support native decode anyway, so I don't personally find it worth it yet.

Just to mention, your Mac (if its an M series) can handle LLMs reasonably. I have an M2 Mac mini that runs some models, but my web interface for accessing those models is hosted on an LXC on one of my proxmox nodes.

It mostly gets used for local audio transcription and summaries, though occasional I'll let it do other things like autocompletes for code (but not generating code, I don't consider generated code to be useful).