A car is useful to explore the different beaches around the island. Keep in mind that the Big Island, Hawaii is a little more remote than the mainland, so you may have trouble finding Ubers if you venture outside of Kona Town. However, if you think you'll be staying in Kona Town the whole trip, there are all the necessities within reasonable walking distance.
Yes, if you are the adventurous type then there are a bunch of excursions that require a 4 wheel drive car (see the "things to do" section). We've seen many cars towed out from these areas and if you don't want to be that guy we recommend a Jeep Wrangler or Toyota Tacoma ;).
Aside from your spiffy wedding outfit, there will also be a welcome party that will be Hawaiian casual. Bring your favorite Hawaiian shirt! Temperature is around 75 degrees this time of year. Don't forget sunscreen, a sun shirt (the sun can get a little spicy midday), hiking shoes (if you plan on hiking), swimsuit, light poncho or rain jacket (we sometimes get a sprinkle midday), sunglasses, and your favorite bucket hat.
Kona International! DO NOT FLY INTO HILO. Hilo is about a 6 hour drive from Kona and is considered the local part of the island.
Yes. There will be shuttle services available.