Yes, the access point works with *any* router. Connect router and access point with Ethernet cable and then configure the access point (e.g. with the iOS app, or UniFi Cloud Key, or UniFi Cloud).

Important: If the router itself does not supply power (Power-over-Ethernet PoE) via the Ethernet port (unfortunately this is the case with the ER-X), you still need to loop the Ethernet cable through an Ubiquiti power adapter (should be included in the scope of delivery of the AP-AC-M, if I interpret this attribute correctly: "May be equipped with a power adapter: Yes").

Yes, it works, as has already been written correctly.
But in order to set up these UniFi APs correctly, you need the Unifi Controller. Configure once and then the controller is actually no longer necessary. Not part of your question, and you don't say anything about it, so I don't know your level of knowledge. I just want to mention it, because otherwise nothing that has to do with WLAN will work, because the WLANs have to be set up in the controller (SSID, password, any VLAN, finetuning for transmit power, channel, etc.).

Yes, it works. In this case, the device would be used as a pure AP.
If the coverage in the outdoor area is not sufficient, several of them can be used, which then expand the network in a self-organising way, without each access point having to be connected to a LAN. Mesh just... Then each access point would only need power.
I hope this helps with the question.