What's the brand and model of the router?
> 1) if router has connected to my WISP (it should connect to it without being authenitcated, right)?
Although most PPPoE implementations use authentication, it's not actually part of the specification.
> 2) has been authenticated by my WISP?
Sometimes on the WAN Status page it will say whether it's connected or not. Hard to say how to tell that without knowing what router it is you have.
But let's back up for a second:
> When I set wireless router in AP client mode
Usually that mode means it talks only to one particular (wireless) WAN connection through the radio (and none locally). The only connection[s] it will talk to locally in that mode would be through the wired LAN ports. Again, without knowing what brand/model the router is, there's no way to know for sure... each manufacturer tends to name things a little differently so you'll buy more components made by them to ensure they'll all work together.
