Plum smacked it right on the head.
I can't tell you what a bad ... very very bad ... idea it is to run power directly from the battery to the head unit. Always wire from the cabin accessory.
Those pictures bring back some memories though. I learned the hard way by just taking the cutters to the harness in my first "install". When you're 15 years old ... you know it all until you figure out that you don't. After that, whenever someone brought me their car ... it was almost always the same situation and I would tell them to bring me the original plug ... and then slap them on the back of the head.
I would find an original stereo connector for your car and follow the diagram Plum posted. Take everything back to stock. From there you just pick up an adapter on-line or at pretty much any stereo/electronic store or big box chain that sells car stereos.
I would however go with beefier speaker leads.
As for fusing, that's what running off accessory is for. Any head unit you buy today will also come with both an internal fuse and an in-line fuse on the hot wire. About the only thing you'll need to run ... and this is only if you're going with an electronic deck as opposed to an original analog deck ... is a continuous power line. This can also be wired into the cabin fuse box. You'll need it in order to keep your presets, time ... and all that. It would also be a good idea, should you do that, to add an inline fuse to that specific wire. It looks like they took the lazy way and just a continuous line into the most accessible hot wire they could find instead of just taking 2 minutes and some zip ties and doing it right.
Good luck ... because that looked like a mess begging for a fire.