Why does Qatar use a single IP address when it was given more than 800000 IPs by IANA?
At first glance, this doesn't seem to be true – according to WHOIS, 188.8.131.52 is part of a larger "Mobile-Broadband-Pool-No-6" having ~90 addresses. So maybe it's only a slight exaggeration – many mobile ISPs in other countries also put thousands of customers behind a tiny address pool.
(However, note that the Wikipedia userpage was created in 2009 – it's quite possible that the address could have been repurposed since then.)
A reverse search for "mnt-by: QTEL-NOC" in RIPE's WHOIS server shows a large amount of IP blocks used for various purposes – there are FTTH pools, schools, point-to-point links, each internet café has its own inetnum entry. There are even some IPv6 allocations (2001:1a10::/32).
This seems detailed enough to be real, but at the same time detailed enough to be suspicious... But many of those addresses are actually routed via AS8781, AS42298 (Ooredoo, the Qatar ISP) and AS198499 (Qatar University).
It could be that some hosts are assigned public IP addresses as usual, but outgoing connections are NATed once they leave the country's network... Yes, NAT works with any address, not necessarily "private" ones. I can't imagine why though.
The reverse search can be done using
whois -r -i mnt-by QTEL-NOC, optionally with
-T to request only specific records such as "route" or "inetnum". However, be careful with reverse searches as the WHOIS server may temporarily block you very quickly due to requesting massive amounts of results.
Did they sell all other IPs to someone, or do they use these IPs for something non-public?
As a side note, some other places actually do that – for example, the US Dept. of Defense owns the address range
184.108.40.206/8, but none of it is routable from the public Internet – it's only used for their private networks.
With a single public IP, isn't the proxy server limited to 65536 simultaneous connections (by the number of available TCP ports)? How do they deal with this limitation? Or is the whole country limited to 65536 simultaneous Skype calls?
Each TCP connection is identified by two IP addresses and two ports. Just like you can open multiple connections to the same host because they use different source (or destination) ports, the opposite is also perfectly valid – connections to different hosts may use the same ports, as long as at least one value from (src-ip, dst-ip, src-port, dst-port) differs.
So if the gateway is sufficiently smart, in total it could handle 65535 connections towards each host:port. (Port zero is not used since it causes problems in many systems.)