But what's going to happen if I visit the LAN IP address assigned by the network administrator, compared to visiting localhost?
Will it go ask the router/gateway about who e.g. 126.96.36.199 is?
No. Generally, the OS will realize that it owns the destination address, and will not send the packets anywhere outside.
For example, on a Linux system, this happens through the routing table – if you look at the
local table, you'll see all assigned IP addreses having /32 routes pointed at the 'lo' interface.
In other OSes such as Windows or BSDs, the mechanism may be more opaque, but the end result is nevertheless the same. (The only exception is when using "jails", "containers", "network namespaces", or "VRFs" – each instance only recognizes its own IP addresses, and not those belonging to other jails/containers.)
Whether the address is "public" or "private" does not matter.
And could the router fool/tell it that 188.8.131.52 is outside the LAN and tell it go out to find it? More precisely, can NAT routes LAN 184.108.40.206 to the public, real 220.127.116.11?
If it exactly matches your computer's IP address – no. The computer won't even contact the router in the first place.
If it's in the same subnet as your computer's IP address – possible, but with difficulties. The router would need Proxy-ARP in addition to NAT. It should be doable with Linux iptables in theory, but I'm not sure if any end-user routers have that feature built in.
I am asking this because I think my ISP gives me a public IP as LAN IP address (less possible that I truly connect to that public IP)
It is normal that ISPs issue public IP addresses, and that's because ISPs do not generally issue LAN IP addresses: that's the job of your router. The ISP only provides initial configuration for that router (and only if the device itself is ISP-provided). The router itself, of course, gets its address from the ISP.
It is possible that the provided device is actually just a bridge (or a router in bridge mode), in which case receiving public IP addresses is completely normal.
That said, some ISPs are known for "borrowing" previously-unused IP space. For example, the range 18.104.22.168/8 sat nearly-unused for many years (until CloudFlare picked it up), so various network admins sometimes decided to use it as an extension of the standard private address ranges. In those situations, best to contact customer service and submit a specific complaint that you're unable to reach some specific service at the overlapped addresses.
(Note that the range 100.64.0.0/10 is not public, quite the opposite. It's a private range meant to be used strictly between the ISP and its customers, in situations where they don't have any public IP address.)