When Shelley uses this word, we're not sure that he literally means a place where God lives with a lot of angels. Christian spirituality definitely isn't a major focus of this poem. At the same time, it's hard not to think of that connection, especially when he capitalizes the word, like he does here. By dropping this word in here, he can connect the beautiful song of the skylark with divine forces, without going into a lot of detail about what exactly he means by that.
Line 3: In this case, he imagines the bird singing from "Heaven, or near it." See how he's a little slippery about the whole heaven thing? The speaker wants us to know that this is a magical creature of the air, but he doesn't want us to get too caught up in details about heaven.
Line 18: Again, heaven is capitalized here, and you might have noticed that Shelley is pretty careful about the words he chooses to capitalize in this poem. This lets us know it's important, and also makes us wonder if he isn't making an allusion to the Christian vision of heaven.
Line 30: In this case, "Heaven" seems like it might be a more generic poetic reference—a fancy way of saying the "sky." Everything in this poem is intense, and there's a little bit of hyperbole here. The sky can't really overflow, but our speaker sure feels like it can.