The language of the song is very poetic and quite unusual, so a lot of it is not standard conversational English. In particular, grammar rules you may have learned at school are often broken in order to achieve a particular effect.
1. I think "blackbird" was chosen simply to keep the rhythm. It could have been almost any common songbird with two syllables, but "blackbird" probably simply sounds the best.
2. The author is comparing the morning with the creation of the world. Normally, plants spring up and grow quite slowly, but here the author is imagining the Creator walking along, and wherever he goes the plants spring up already complete -- already fully-formed.
3. I'm not sure, but I think the author is suggesting that the beautiful dawn is a gift from God.
4. In the Bible, Eden is the name of the garden which God originally created as a beautiful place for humans to live -- a "paradise" (the word is an old word for "garden") where everything is beautiful and there is no pain or suffering. The sentence here is elliptical (for poetic reasons): it means, "The sunlight and the morning are born of the same light which the Garden of Eden saw playing." Light shining through trees when there is a slight breeze can be seen to dance around, almost as if playing. The author is saying that the light of the morning comes from that same light in this paradise.