Upon creation of my resource pack, do I have full ownership (exception of Mojang) for what I want to do with it and what others are allowed to do with it?
First, don't include anything that Mojang made in your pack. They have their own copyright, and unless they say otherwise the default textures aren't fair game. I'm not sure if that's what you meant, but I just wanted you to be aware that your pack will likely be removed from this forum if you include anyone's work (including Mojang's) without their permission.
Aside from that, yes, you automatically gain some protections of copyright for anything that you have created that you publish online. This includes the right to decide how it's used, shared, and so forth. I suggest brushing up on the DMCA if you're living in the US, or the Berne Convention if you're in most of the rest of the civilized world.
As long as you aren't violating someone else's copyright (see above) your stuff is automatically legally copyrighted upon publishing. If you live in the US you can still register with the US copyright office, though. This has the advantage of allowing you to sue for damages if someone steals your stuff. The disadvantage is that it costs like $300, which is probably more than you'll make from your pack. If you're living in a country covered by the Berne Convention, you get this protection automatically... but it's still a good idea to register your copyright so you have proof of ownership.
As to writing a license, yes, you can do that. By default anything you publish is released as "All Rights Reserved", which means that nobody can do ANYTHING with your work... including downloading it. Now... if you publish something you're giving implied consent for people to download and use it, but it's still a good idea to put that in writing. But even with that implied consent, your pack is locked down in terms of redistribution and remixing unless you specifically allow it.
That's where a license comes in. Starting from that "You can't do anything at all with my stuff" frame of reference, a license explains what liberties you want people to have. Do you want people to redistribute your work? Use it in videos? Continue working on it if you suddenly vanish from the internet? That's the stuff that goes into a license. There are plenty of freely available licenses available for you to use, but you can just write your own. As long as it's unambiguous it will work... in so much as anyone cares about the rights of artists these days.
If I don't legally copyright, will others just take my stuff and remix it as they please without me being able to do anything?
Have you been paying attention at any time since the early 90s? People will steal your stuff even if you DO file for a copyright unless you take the time to sue them when they rip you off. And even then they'll probably still do it thinking they won't get caught. Particularly for Minecraft, there are entire sites devoted to making a buck by ripping off resource pack creators. It's the sad reality of this game now that it's become so commercial.
Can someone explain this mess to me? The Eula doesn't get very in depth.
The Minecraft EULA covers Minecraft, not your artwork. For that you need to turn to copyright law. Your art belongs to you, and nobody should be stealing or using it without your permission. If you want to give people permission, you need to release your work under a license be it a self-written list of permissions or a proper legal document. Not that any of this will likely stop anyone from just doing whatever they want (because people don't care about artists' rights), but legit sites like this one will try our best to honor your copyright and license terms which should keep theft to a minimum... except for Youtube which doesn't care unless the copyrighted content is in a video.