Not sure you need an access point or a wireless router. So buy a wireless router because access point cannot be used as router but router can be used as an access point. :D...

You need WISP mode? It doesn't support it...

Detination IP Address is related to Static Route configuration. To set a static IP for xbox, I think you should go to MAC & IP binding and Address reservation.

I think your modem is faulty...

If there is WAN IP, internet connection should be OK. Any other wireless clients connect to it as well? What's the OS of your laptop? Try to ping google,com and ping 8.8.8.8 to see if it is DNS issue.

Try to login the router's web interface when it disconnects. If you cannot, check IP address, default gateway obtained on the laptop. Probably you should upgrade firmware of TL-WR702N. By the way,...