The Apple adapter will not give you full resolution.
Apple HDMI to DVI Adapter - Apple
"Use the HDMI to DVI Adapter to connect a single-link DVI digital monitor, such as the 20-inch or 23-inch Apple Cinema Display.
Supports single-link DVI devices up to 1920-by-1200 resolution."
You can connect a 30" Apple display to one of the Mac mini's Thunderbolt/USB-C ports with USB-C to Dual-Link DVI adapter.
https://www.startech.com/en-us/audio-video-products/cdp2dvidp
You should connect the display's USB plug to the computer. It is used for brightness control. The FireWire plug is not needed. It is just for the FireWire ports on the back of the display.
If you want to use the computer's HDMI port, it gets more complicated. There are no HDMI to Dual-Link DVI adapters. There are DisplayPort to Dual-Link DVI converters, such as:
https://www.startech.com/en-us/audio-video-products/dp2dvid2
that you could use with a HDMI to DisplayPort converter, such as:
http://www.foinnex.com/pd.jsp?id=309