Computer to monitor cable

Use the Thunderbolt cable that shipped with the Studio Display.

see "In the Box" at > Studio Display - Technical Specifications - Apple

For reference, also see > Connect a display to Mac mini - Apple Support

The 27" Apple 5K Studio Display has 5120x2880 pixel resolution, and the default setting for it is probably Retina "like 2560x1440" mode. Text and objects are sized as if the display had 2560x1440 pixels, but most drawing is on a 5120x2880 pixel canvas whose dimensions match the 5120x2880 pixel screen.

I'm guessing that your Mac mini's support for 6K resolution means that you could run that display in Retina "like 3008x1692" mode. In that mode, the drawing canvas would have 6016x3384 pixels (the resolution of a 32" Apple Pro Display XDR), and the Mac would downscale the 6K image to fit onto your 5K screen.

This would give you a little more "workspace" at the cost of smaller text and objects.

Even if you were using a HDMI port that supported an 8K display, that particular display expects Thunderbolt input. There is, as far as I know, no good way to convert HDMI to Thunderbolt.

You must use Thunderbolt to connect that display.

Studio Display (5K) only has 1 Thunderbolt Port - dedicated for connecting with Mac (also needs to support 5K with TB capable). There are 3 USB-c ports for other peripherals.

Thunderbolt connections (in your case, it TB 5) provide video signal and DATA - that's why it's default to use TB cable for connection: therefore, the webcam, speaker, microphone will function, as well a HUB for the 3 USB-c ports.

Just stick with TB cable - even my spouse's Mac Mini M1 is very happy with Studio Display.