There is a big old carousel in the main square which my children loved. There is also a children's park above the Palais du Papes with slides etc. There is lots to see in this area. We didn't find Avignon drab and I think you would be fine staying there although we've always been in nearby villages.
We went to Avignon on Eurostar. Journey was pleasant and easy, but be careful which seats you book - we ended up in some weird little cramped cabin at the end of a carriage with a baby whose mum never changed his pooey nappy once! Avignon itself I think is a bit drab. There's a big castle which has nothing in it, and souvenir shops. There's the famous bridge of course. Arles is prettier, has coliseum and Van Gogh stuff, also a great market. I LOVED Aix en Provence, so pretty, great shops and markets. Orange is fun for the roman amphitheatre but only worth a day trip. Marseilles would be good for a trip to the coast.
We have stayed in Arles twice. Eurostar direct from St Pancras to Avignon, then hired a car for Arles which was about 45 mins away. Anyway we love the area, loads to do on a city break. Avignon, like Arles, very historic if that suits you, and just wandering, nice cafes etc. First time we went dc were 3 and 9, then 2nd time 5 and 11. They enjoyed it but they are city kids so are used to hard pavements (!), museums, shops rather than wide open spaces. If you are planning on staying in Avignon then I see no reason why you would need a car. We used the bus there and it was pretty straight forward iirc. I really love that part of France and will o doubt return.
Oh and one thing. When you are booking Eurostar, just look and see ow much more it is to go business class. We paid something minimal extra, like another £30 per person and for that had 2 meals served to us, tea, coffee, alcohol I think, more space. For a 6 hour journey it was well worth the not too much extra cost.
Am I crazy to consider a week in a city apartment there in June with a 4 and 6 year old?
I figure there are outdoor pools and we could do some gentle sightseeing plus maybe a day at the coast. I am also considering doing Eurostar and not hiring a car - using public transport/walking when we get there.
I've never been there - please advise! Thanks a lot.