1. That sub alone is ~$200. I think the stealthbox is like $700? so you're paying $500 for the box?! I'm not sure how much a local shop would make a box, though if you have some woodworking skills try making your own it isn't that hard with the right tools.
2. You don't NEED a class D, however they are very efficient and run cooler as well I highly recommend it. Try to get one that is rated around '600W RMS @ 4 ohms' a little higher is fine. The amps will probably have different ratings depending on the ohms. Make sure you look at the rating at 4 ohms since that is the rating of your sub.
I'm not really familiar with the shaker 500 amp for your 5x7s so I can't help you there. Personally I run an amp on my components.
3. I'm not sure if the 2013 has rca outputs on the stock HU. If it doesn't you'll need a line out converter. It allows you to connect the speaker level signal from speaker wire into rcas so you can hook it up to your amp.
As far as wiring goes you can just get a 4 gauge amp kit. HOWEVER, not all amp kits are equal. A lot of cheap kits advertise 4 gauge, but when you strip the insulation it reveals something closer to 8 gauge.
Also look for OFC (oxygen free copper) instead of CCA (copper clad aluminum) as they conduct electricity much better although will be slightly more expensive.
If you want to buy the parts separately you'll need:
4 gauge wire - Not sure how long you need but you can get 25ft to be on the safe side. I think I used ~18 ft for the power and 1 ft for ground. Shorter the better.
Fuse - place it as close to the battery as possible
Ring Terminals - for the 4 gauge
Speaker wires - for sub ( if it's not included with stealth box)
RCA cables - connect line out converter to amp.
Remote Turn on wire - simply 18 awg stranded wire it turns the amp on when the headunit powers on.
I might be missing some other stuff you can always look at a pre-made amp kit as a check list lol.
08 V6 Mustang
JLT CAI, 93 OCT race tune, pypes x-pipe w/ gt take offs, 8.8" w/ 3.73, shaftmasters 3.5" aluminum driveshaft, gt front/rear swaybars.