You do not appear to understand what you're asking.
Public IP Address
With the rent of domain from the registrar, I automatically get a static public IP address?
No, that's not how it works.
If I translate your question to the real world, you are asking for a postal code and a door too, but you don't want a house.
A postal code can be linked to the door, yes. But you cannot get a door without the rest. The door is part of the house the same way an IP Address is part of an internet connection.
When you rent webspace, you are given an IP Address. When you get internet access at home, you are given an IP Address.
Yes, in some places it is possible to get a 2nd IP Address the same way it is possible to add a 2nd door to your house, but without the house and the first door, the 2nd door is as meaningless as the extra IP Address.
So in order to get a public IP Address that is static, you need to have something attached to it. The IP Address is only a connection point to an Internet service. If you want a static IP Address to connect to your own internet, then you need to buy a static IP Address from your ISP.
Email hosting
You also talk about email, this requires an email server. Email hosting would give you an ip address, but they often also come with the ability to host websites. So basically, you really do want a website online, but you can choose to not use it.
A simple index.php as only page with the following content:
<?php
header("HTTP/1.0 404 Not Found");
?>
will make it seem the website does not exist.
Alternatively you can use services such as Office365 or gmail. You then don't even need a public IP Address. All you do is set the MX record to your office365 account: eg: mydomain-ext.onmicrosoft.com or link the domain through gmail.
DNS
The Domain name is completely separate from all this.
When you buy webhosting services, they often include a free domain name, because you need it anyway, and it is a marketing strategy because not every webhosting service gives you a free domainname too (most do nowadays, but not all).
The domainname is configured using DNS and all it does is link a name to an IP Address and have some additional settings.