Are grills at the shelters

Yes most that I've seen have a grill, brick oven, or for pit with grill

Yes. You will probably want to bring something to clean it before use however.

Yes

There are multiple grills... Some are at the shelters, some are next to random picnic tables, some are under pavilions. They are relatively well maintained, and are easily adjustable for height and to pour your coals in. I suggest that you use a chimney to start your coals however because charcoal tends to take a long time to heat up in this style of grill. Make sure to bring a wire brush, and tin foil isn't required.

Yes they all have a grill

Yep