If only. An IP address is issued by Nominet which is publicly searchable. That said, its possible (or was, at any rate, when i first registered in 2010) to request ex-directory status similar to old BT landine regs.
Not quite
Nominet manage name web addresses not numbered IP addresses
The nominet Whois database identifies the registered owner of an address, and that can be anonymised.
For example you could identify either the individual/corporate owner of a website, or just their hosting provider
Your IP address/addresses is/are issued by your network provider. Probably a unique master address to your router and a localized address from your router to your device.
Check whatismyipaddress.com to identify the unique Internet address allocated by your provider
The use a Whois search to identify that. You will find the details of your Internet Provider not of you personally