Second week in September: HalloweenTime may have started, which means Haunted Mansion and Space will (probably) have their Halloween overlays in place. (They haven't announced the start dates of HalloweenTime yet; that usually happens in mid-summer.) The biggest drawback is that HalloweenTime is very popular, which means weekends are usually very busy in the parks. Hopefully, by then GotG (ToT) will have re-opened, as will have F!, the RoA, and the train.
First Week in May: This is before the summer season starts and after Spring Break season has ended. ToT, RoA, F!, and the train will all still be down at this point and there is the potential for other attractions to also experience closures (with or without much notice). I think there is a chance the crowds will be lower here (but DLR has not announced if they'll have any special activities - like AP Days - during this period yet).
Weather: I don't think you can count on good weather for either period. I've been there in May and September when it's been hot and when it's been nice!
Schedule: Both of these periods are nominally considered non-peak season, so expect shorter operating hours and reduced entertainment (e.g., fireworks on weekends only).