Japanese almost certainly won't be eating on the bus, but I don't see any reason why you shouldn't have snacks, especially if you have small kids. I've had a sandwich or onigiri on the bus before, and it will not be a problem. Drinks are fine too. I've never heard any announcements or seen anything saying that you can't eat, and certainly the Access Narita website doesn't say you can't. One thing to remember is the luggage size restriction, max one piece of luggage each up to 20 kg in the luggage compartment for Access Narita; effectively no restriction on the Limousine Bus. But since you're planning to take a train from Ginza or Tokyo onwards, I assume you won't have so much heavy luggage anyway.
From memory, I'm pretty sure all the buses have toilets, but, as you say, Access Narita buses do.