If you've got a relatively recent gpu, NVIDIA is fine on Linux these days. It's changed a lot in the last two years specifically.

HDR works fine outside of games, but is limited on the number of Wayland compositors that support it. KDE plasma and Gnome are the only two I know of. Most of the others don't.

HDR can work with games, but it involves a bit more playing around under the hood with gamescope. But that's an issue with steam/wine, that impacts NVIDIA and AMD.

I've been running an NVIDIA setup with two HDR monitors on KDE for a couple of years now, and it's perfectly fine these days. It wasn't when I started :)

1. I run an Nvidia GPU daily, with little to no issues. I can stream via OBS, I can play nearly all of my games (with very, very few exceptions), and I can still use features like raytracing. Note that I don't do anything with CUDA, so I can speak to that if that's something you plan to utilize.
2. AMD has a more open driver implementation, making it way easier for Linux devs to integrate and improve things for the end user. There's currently an open source community project called NVK that is likely to supersede the closed-source Nvidia drivers, but that's still a few years away from being fully at parity (though it's making big strides relatively quickly).
3. I don't have HDR, so I do not know. My vague memory indicates support stems from Window Manager support (i.e. the engine that draws windows and determines their behavior), so look into which window managers implement HDR.

I have an Nvidia GPU and I’ve been able to get HDR working on Linux (Bazzite gnome) with some tweaking for specific games. I haven’t really tried watching movies with it yet.

1. Yes. Nvidia drivers are usable enough, but they regularly cause power management related issues for me. I gave up on sleep because of Nvidia drivers. One update breaks it and then the next update fixes it, only for the next update after that to break it again. Never had any issues before the current Nvidia GPU.
2. Their drivers are simply better integrated into the OS and better developed for Linux. That's all really. AMD drivers have more effort put into them.
3. Bad. Not good. Expect tons of compatibility issues and straight up impossibilities and jumping through hoops.

My 3070 worked fine with hdr in kde arch for over a year.

I will say though HDR isn't something you should turn on unless you have a display that does good HDR, otherwise everything looks washed out.