I've a HP ProDesk 400 G3 Mini (1EX82EA) with a 1 M.2 2280 slot for an optional Wireless NIC.
Can an Intel® Dual Band Wireless-AC be installed in this type of computer? And which of the models will be compatible?
Intel® Dual Band Wireless-AC 7265
Intel® Dual Band Wireless-AC 8265
Or any other type...
Many (in my opinion, slimy) manufacturers will implement whitelists in their BIOS. These whitelists list the parts that will be allowed to operate - thereby forcing you to purchase your add-on hardware only from this manufacturer and at their inflated prices. Now, I am not saying that this is the case with HP (I really don't know); what I am saying is that you should be asking HP for the answer to your first question.
Presuming that you have a A/E-Keyed M.2 connector in this PC and that you can use any compatible WiFi adapter, your choices from Intel include:
https://www.amazon.com/Intel-3165NGW-Wireless-Bluetooth4-0-802-11AC/dp/B0194GE5YC Amazon.com: Intel 3165 Dual-Band Wireless-AC + Bluetooth 4.0
https://www.amazon.com/Intel-Wireless-Generation-802-11AC-Bluetooth/dp/B071JR1YN5 Amazon.com: Intel 3168 Dual-Band Wireless-AC + Bluetooth 4.2
https://www.amazon.com/Intel-Wireless-AC-802-11ac-Wi-Fi-Bluetooth/dp/B00STV5UKW Amazon.com: Intel 7265 Dual-Band Wireless-AC + Bluetooth 4.0
https://www.amazon.com/Wireless-8260AC-8260NGW-Bluetooth-Wireless/dp/B01ERMBDCK Amazon.com: Intel 8260 Dual-Band Wireless-AC + Bluetooth 4.2
https://www.amazon.com/Intel-Wireless-AC-8265-NGWMG-BlueTooth-Brown/dp/B01MZA1AB2 Amazon.com: Intel 8265 Dual-Band Wireless-AC + Bluetooth 4.2
Hope this helps,