That's incredibly confusing on the Airport; it doesn't show you enough information like what the actual LAN IP address is! We can guess that it must be 10.0.1.1 but it doesn't actually say that anywhere...
The WAN settings on the Airport are inconsistent. It is configured to use the private LAN IP of the 2wire for DNS but the public IP as the gateway.
There are a few options with your setup, and the Airport seems to be trying to use half of one and half of another... It's no surprise that you can only access some parts of the internet. What is a surprise is that it's working as well as you say it is!
Option 1: Clever 2wire, Stupid AirportThis is what I would do.
Turn off DHCP on the Airport and assign it a static IP of 172.16.0.254 / 255.255.0.0 with a gateway and DNS of 172.16.0.1
The 2wire is already set up with a private IP of 172.16.0.1 and NAT turned on, but you need to tick the "Enable DHCP"
Make sure the cable from the 2wire is going into a LAN port on the Airport; and not into the WAN port
The 2wire now does everything; the Airport is acting as a Wireless Access Point and a switch. It is not acting as a router.
Your computers should now pick up IP addresses of 172.16.0.33 upwards.
If you require Airport services to be accessible from the Internet, you will need to forward the relevant ports on the 2wire.
Option 2: Clever Airport, Stupid 2wireTurn off NAT on the 2wire; Tick "Bridge Network". It is now not really acting much like a router; the Airport will use a public IP.
Make sure the cable from the 2wire is going into the WAN port on the Airport; and not into a LAN port (opposite of Option 1)
On the Airport, I'm going to guess the displayed WAN IPs must be correct for it to be working at all:
Configure IPv4 - MANUAL or STATIC
IP Address: 212.139.91.190
Subnet Mask: 255.255.255.252
Router Address: 212.139.91.189
DNS: 8.8.8.8 and 8.8.4.4
Leave everything else as it is; DHCP and NAT enabled.
The computers should pick up I{ addresses of 10.0.1.2 upwards.
The Airport should be accessible on your public IP address.
Option 3: Clever everything (double-NAT)This is what most non-technical people do, because it's the way it comes "out of the box" and it should "just work".
Enable DHCP and NAT on both boxes
Plug the Airport's WAN into the 2wire.
The reason I would not do this is because it's translating the network address twice, which doubles the chances of a problem occurring as well as introducing unnecessary overheads.
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