What are the specs I should focus on for late-game Astrals ? Healing, Caster, Melee ?
What are the main healing spells right now ?
Late game astral, you're almost assuredly going to be in healing aspect. You may luck out and find yourself in a party where someone else is doing the healing, but the chances are fairly slim. In general, though, I think caster spec is used for astral DPS, with melee spec used for PvP in Support aspect. It is entirely possible that melee could work just fine for astral DPS, I cannot say that I know for certain, but my personal experience with trying to use caster spec DPS was that it was rather difficult to do right. It was likely less a matter of what talents and rubies I had and more about knowing the proper rotations, but I found my DPS at the time to be subpar. Regardless, you'll likely be healing.
As far as primary healing spells... Miraculous Healing and Halo are really your two main heals. Touch of Light is used for party-wide healing when people are down a bit, but not in critical danger, or if they're spread out. Healing Light is a kinda crappy spam heal that is mostly for gaining back Faith/Fanat or when you need to heal and have no Fanat to work with. As much as possible, though, you're going to want to try to keep the barrier from Holy Word up on the tank. Plate healers are as much about shields and getting the Protection buff on the tank as they are about actually healing. Wardens are the ones to go with if you just want to do All The Healing Ever.