I think Kodi does some casting... Not 100% sure.

Make sure TearFree is enabled in the graphics configuration (google it).

The hardest part in my project was recalibrating the colors because my TV in HDMI-RGB mode (as opposed to YCbCr) displays everything below 10% brightness as black. The rest was done very quickly. I don't even need a sequential-chunk torrent software because I use FreeRapid Downloader and ulozto.net (can download fast enough to play while downloading at reasonable 720p/1080p bitrates, otherwise ulozto-downloader and a 10min wait is needed).

Kodi supports DLNA, a media library sharing/casting standard.

You can use a relatively cheap Pi as a NAS (network-attached storage - there are ready-made solutions but expensive and don't get updates for nearly as long), and possibly add Kodi media server capabilities (useful for smart TVs). Check if that model supports a sufficiently fast disk interface (USB 3.0, SATA etc.) and Ethernet (100 Mb/s or better if 4K is required). Boot from an SD card and use a 2TB+ HDD (1TB could be enough if you barely store anything). Most disks from the past 10 years will be good enough to play 4K video from if no OS is running from them. Go for a lower-end one but not ridiculously cheap, and check that people aren’t frequently complaining about the model or product line.

I don't know which Pi models can smoothly play HD video without overheating, I don't own any. But those that can are likely a lot more expensive than old PCs you could use otherwise. I would just get a cheap one for a NAS and probably some other common network use cases (web server, Pihole etc.)

Yes just research how to install Plex or Kodi on your pi. I just did it this evening for an update. For that I used docker with Linuxserver/Plex. It takes some time to get all the settings but there are good YouTube tutorials on how to do it.

You'd be better off with a used office pc, something with a 4. Or 6. gen Intel CPU is usually cheaper than a RPI and way more versatile. Only thing you lose on is size and power consumption.

This may seem like a bit much, but it pretty much holds your hand to guide you through the install.

Have you tried Kodi?

Is jellyfin a better alternative?