If I register a .ie domain, does that prevent others from using similar names as .ie domains, or even the same name as a .com, .org etc?

No, if you register a domain name this does not prevent other people from registering similar domain names or variations. If you do not want other individuals or organisations to be able to use variations of your domain name, you should register them yourself. For other Top Level Domains (TLDs), someone can register the same .com, .net, .org etc. domain name as your .ie domain name unless you have already registered it yourself.