Leased IP Blocks
IPs are assigned in blocks by IANA to the Regional Internet Registries (RIR). See this (list and map) of the RIRs. The RIRs then lease out smaller blocks IPs to individual companies (usually ISPs). There are requirements (including fees and proof of use) for getting a distribution and failing to maintain these means a loss of lease.
Once a company has leased one or more blocks from the RIR, they need some way of telling the rest of the world where to find a particular IP (or set thereof: subnets). This is where BGP comes into play. BGP uses a large network concept called an Autonomous System (AS). The AS knows how to route within itself. When routing to another network it only knows about AS Gateways and where the "next hop" toward those external addresses. AS numbers are managed by IANA as well.
Within an AS, even one as large as an ISP, they might use several routing protocols (RIP, OSPF, BGP, EIGRP, and ISIS come to mind) to route traffic internally. It's also possible to use Static Routing Tables, but entirely impractical in most applications. Internal routing protocols are a huge topic, so I'll simplify by saying there are other questions on Server Fault that can do those topics more justice than I can here.
DNS
Humans don't remember numbers well, so we invented host names. Skipping the history, we use the Domain Naming System (DNS) to keep track of what hostname points to what IP address. There is a central registry for these, also managed by IANA, and they determine what Top Level Domains (TLD) (eg ".com" or ".net") go in the Root Zone, which is served by the Root Servers. IANA delegates administration of the "root zone", this administrator only accepts updates from qualified Registrars.
You can use a Registrar to "purchase" a domain name, which is a subdomain of a TLD. This registration essentially creates that subdomain and assigns you control over it's Name Server (NS) and Glue (A) records. You point these to a DNS server that hosts your domain. When a client wants to resolve your IP from a domain name, the client contacts their DNS server which does a recursive lookup, starting with the root server, finding your DNS server and eventually getting the relevant information.
Everyone Agrees
As for the "governing bodies": everyone just agrees to use them. There are no (or very few) laws requiring anyone to cooperate at all. The Internet works because people choose to cooperate. The governing bodies provide a means of easy cooperation. All the various RFCs, "Standards", and such - nobody is being forced to use them. But we understand that society is built on cooperation, and it's in our own self interests to do so.
The efficiency bred by cooperation is the same reason BGP is popular, everyone basically agrees to use it. In the days of ArpaNet they started with hand configured route tables; then gradually progressed to a more comprehensive system as the Internet grew in complexity, but everyone just "agreed" to use whatever new standard. Similarly name resolution stated with host files that networks would distribute, and eventually grew into the DNS system we know today. ("Agreed" in quotes because many times a minority set a requirement for a new standard and nobody else had a better alternative, so it was accepted).
Trust
This level of cooperation requires trusting IANA, a lot. As you've seen they manage most of the various systems' cores. IANA is currently a US Government sponsored Non-Profit corporation (similar to the US Post Office), it is not part of the government, though only barely removed. In past years there was concern that the US Goernment might exercise some control over IANA as a "weapon" against other world governments or civilians (particularly through laws like SOPA and PIPA, which were not passed, but may be the basis for future laws).
Currently IANA has taken it upon themselves to raise funding (despite being a non-profit company) through the creation of new TLDs. The "xxx" TLD was viewed by some as an extortionist-style fundraising campaign, as a large percentage of registrants were "defending" their name. IANA has also taken applications for privately owned TLDs (at $180,000 each; they have suspended the application process after being inundated with applications, nearly half being from Amazon alone. Many of these applications resulted in new gTLDs.
I think you're overthinking this design and going about it the wrong way.
PAT1, PAT2, and LB1 should all iBGP peer with one another.
LB1 should originate 199.192.100.0/28
PAT1 and PAT2 should both originate 199.192.100.0/24
Do not put any filters on any of the ibgp sessions.
What IGP (OSPF?) are you running between these routers? The default route PAT1 and PAT2 are originating should come from the IGP, not from iBGP.
The problem isn't that PAT1 isn't announcing a backup path to the /28, the problem is that PAT2 doesn't believe it can get to LB1 via PAT1. What does a 'show ip route' on PAT2 for LB1 look like when the link between PAT2 and LB1 is down? If you take down that link, the iBGP session between LB1 and PAT2 should still stay up, if it goes down, again, that points to an IGP problem.
Best Answer
It works fine. It just requires a lot more engineering work to make it go, you need routers, BGP connectivity, your own IP space, etc. I assume in your example above that you have data center 1 and 2 running all the time.
Many people do this fine, look up 'anycast' for what you're trying to do. The large problem with this is that it works much better for UDP based services (non-stateful). If you're downloading a large file via HTTP, and there's an outage on the router side, your traffic will go to the new 1.1.1.1, which has no idea what is going on, and will drop the connection.