Will a website be able to block my IP address alone somehow, while I am running a security tool behind my NAT? Or will it block my ISPs gateway IP?
The website has no way to know your intranet IP. Although it could be possible, it would require some extra information which is part of the app-level protocol. For example, the skype servers know your localnet ip, because your client says it to them. But the ordinary http(s) doesn't contain this.
The common behavior in such cases that the websites simply aren't interested in your localnet address and if there is an ip-level block, they block your gateway IP - and your whole localnet with it.
But not this is the only possibility. Some websites - the most common is the wikipedia - use a more complex user identification using cookies, and - if you log in - your login name. This enables them to re-activate or extend the to you belonging IP blocklist if you would try to login them from a different (outer) IP address.
It would be also possible to use some browser add-ons (ActiveX, flash) to help the websites to know your original address, although these are highly unpopular today - mainly on security reasons.
Since you are behind a NAT, your local IP is translated by the gateway. So a server/website will get a traffic from your ISP IP, and if they decide to block an IP, it will be of your ISPs. There is no way to block your local IP on a server side.