I don't have an upgraded alt. I use the factory system, factory grounds, 4 gauge wire, 5F cap, etc... with no problems.
It's hard to get pictures of colored lights, neons. I'm sure that looks sweet at night.

You cut the Alpine logo into the cover behind the seat? That looks nice.
Don't take this the wrong way, but I think you are worrying too much. Nothing wrong with trying to take care of your equipment / car. It's psychological the benefit gains from things like the "big 3." You feel better because you did something that in theory you felt to be better. Your brain looks for some improvement, and you notice things that you normally didn't because you're now paying more attention than previously.
Factory electrical will be able to handle a 1000w sub. And your Alpine and 4 gauge can handle it. Ignore "peak" ratings on audio, that's in extremely ideal conditions (which is most likely not happening). RMS is the real life peak, and it will most likely run under that RMS most of the time. You still need to take rms into account, making sure the car can produce that RMS for at least a few milliseconds-seconds. Elantra can do that for 1000w.
It could definitely handle 1000w with a cap. Many people will say capacitors are "band-aids." But a properly sized cap that's not constantly drained completely will help and relieve the battery of stress.
My amp is usually going to peak at 45A at the volume I listen to it at. But on average over the time span of listening it pulls 9A. (There goes my bragging rights, my TOL MTX sub is only operating at 130W) A capacitor can provide that 45A for the fraction of a second/seconds it's needed and then the Elantra's electrical can easily provide 9A. I run my amp on a 30A fuse with the cap the amp is capable of pulling 50A, the cap prevents that 50A from coming from the battery. 1F = 1As/V 1 Farad = 1 Amp second / Volts. Most caps are 16Vs. RMS peak of mine is 60A. 60A/16V = 3.75F. My cap beeps when it's fully drained, I only hear this when running the sub with the car not running and the battery is nearing depletion.
Your Alpine at 1000w would produce simular needs. With you're rear battery, you've already got some padding from stressing the car's electrical. Looks like a stinger so it should be a good one for audio. Caps are faster at charge/discharge and lighter than SLA batteries. But you're perfectly fine with what you have. At 28A you're peaking at about 400w and probably averaging 5-7A.
Avg Ampere usage:
Wire runs from engine to trunk are between 10-15'.
Any decent amp has thermal overload shutdown, and your amp definitely has it. (From Alpine - Current Protection: Over-Current, Over-Voltage and Thermal Protection)
Cars get extremely hot and amps are designed to take that heat. Has your amp ever cut off on you? If not you don't need extra cooling. The entire chasis of the amp is a heat speader for those tiny black FETs you can see in the picture of your opened amp. The amp chasis has a lot of surface area to absorb the heat from them. They don't really give you any decent fins to move air pasted to lower the temperature. And you have to cool the entire amp chasis for it to be effective since the chasis is the heat sink.
You need some fins like this for the fans to be effective:
Big 3, adding grounds, etc (whatever you wanna call it) is more problematic than beneficial. It introduces the potential for ground loops. Ground loops are very bad for audio. The chasis of the car is the a greater conductor than any wire you could ever purchase. The problem is not the chasis being a good ground, it's your connections to the chasis. Make sure there is no paint, connectors are tight, use bolts not self-tapping screws, etc... As long as the connection to the chasis is good and solid, you're fine. Any ground upgrade should ONLY be a larger ground strap from the battery to the chasis. Ground will travel the path of least resistance, it doesn't care what it's flowing through. Giving it multiple paths to flow through means you can't control or prevent it from traveling through a conductor that has noise such as the engine.
The best way to illustrate the potential grounding problems by adding ground wires is in the image below. The engine is already sufficiently grounded, adding a second ground allows power to flow through the engine. The blue line shows how the amp could use the engine as a ground to the battery. If that extra ground was never added, this issue wouldn't have a possibility of occurring.