There is plenty of parking at Queenston Heights. Enter the park from the traffic circle on the Niagara River Parkway. After a short entry lane turn left.

No, sorry they are not allowed.

I recommend bringing your own grill and arrive early enough to claim a picnic table - there are lots of tables, but not all are located ideally. There are covered pavilions, but often large groups will book these in advance. There are decent public washrooms and a small concession stand. There is also a wading pool and a very good children's play area. All free, except for a walk up Brock's Monument - named after the British General who was mortally wounded on this spot during the War of 1812, it is a London spiral staircase to the top where fallout views down river are your reward for this rather confining climb.

I think these good folks have answered your question for you. Have a good time.