Do I have to register my websites name?
I have an ABN but I don't know if I have to register the name of my website as well?
No. Just need to secure it as a domain name. If you do trade as that Website then yes you must register it under your ABN. Hope that makes sense.
If you own a URL you already have IP protection by default. There is no need to register it anywhere. The only web domain safeguards I recommend are to also register easily misspelt variations of your site's name plus additional extensions like "dot your business type" and .net when you own .com.