It's OK to connect a smart bulb to a different WAP to the one Alexa is connected to. What's the security type of your Outside WAP? Only WEP and WPA-PSK password are supported. And you may forget the...

I think the camera uses a bit data to maintain function.

you can set different VLAN ID to divide them

there is no wireless schedule function for this device

If the HS100 wifi light turns green that means it has connected to your router. At this moment just log out and log in kasa again then wait a minute and refresh the device list, it should be...

http://www.tp-link.com/en/faq-944.html for reference. Did you try to disable the kasa skill and then enable?