No, you are going to have to be at a much higher price point for this feature.

I used Ethernet cable works great

PoE will pass through this switch but no it does not generate PoE (power over ethernet)

This does not support PoE. You need a PoE switch for that and at least 5e cabling. You could possibly get a power injector for the device you are trying to use. Most sell one or you can get a generic one just make sure what your device needs. PoE or PoE+

No this is not a POE switch.

yes

You would want your router to be the first connection after your modem. otherwise any device you connection would be on it's own with no firewall. Also your ISP may limit the number of devices or IP's the modem gives out.

Well this really depends. If your modem and your router are the same device, then you are essentially adding additional wired ports. so I am unsure where you router is plugged in. I have this plugged into my modem. The have two routers plugged into this. Thus creating two separate networks.

I would put the switch behind your router and plug your AP into that. You want it behind the firewall of your router.

Ah you mean MYAXIS POINT. Yes but why? Only reason would be that you that they both expect to connect to an intermediate device. You might be better off with what is called a 'cross over cable' with that you can connect two computers directly together. Or your modem and you point of sale device. If you can't get a cross over cable then the switch will work.
Background:
Yes but you could NORMALLY connect only one device. Your modem may have a built in router (my old DSL modem did). The function of home routers is to take a single IP address that the world knows about and map that to multiple IP addresses, usually beginning with '192.168.1.' (also known as Network Address Translation). If your connection to the Internet expects to see multiple IP addresses on the side connected to the switch then you don't need a LOCAL router, there is a router external to the modem, but I've never seen that in a home situation. At work we have 128 IP addresses passed to us but that is a commercial connection. My cable modem at home just send over one. The switch just knows about IP addresses that connect to each of its ports, it doesn't know who assigned them.