No - barbecues are not allowed.

There is a childrens playground at the northern end of the park. I don't know if there are baby swings there though. To reach the playground follow A21 towards Hastings and then turn left into Upper Park Road. The playground is a few hundred yards down on the left. Hope this helps

Best to message them on their Facebook page

There are no dedicated BBQ areas within the park but there are three lawn areas within the first area of the park. I have in the past seen families using disposable BBQs on these lawn areas and have not seen any signs that say you shouldn't. I am not aware of any problems or reports of problems using these BBQs and provided you apply the normal safety rules regarding portable BBQs I can't see any reason why you shouldn't enjoy your BBQ/picnic in our beautiful park.

Unfortunately there is no shuttle service from Heathrow. You can get a service into London where you can pick up a direct train service from London Charing Cross to Hastings. Alternatively there is a train service from London Victoria to Hastings but this usually involves changing trains mid journey.

The old town of Hastings nestles in a valley the hill on each side would require walking up fairly steep grounds and I would not recommend that you try to make the walk. The route following the sea front to the centre of the newest part of Hastings is flat and easy walking. From the town centre to the park is a long and gentle incline that should easily be within your husbands capabilities. The total distance from the old town to the park is probably not in excess of a mile and certainly not more than a mile and a quarter.

Yes at the far end lake

Almost all of the roads around the park have free parking.