As for the accuracy of your poem... Here's are the problems...
1. Good Chinese poetry are indirect. So even though you are conjecturing that it is a forest because of the word "green", it is better to keep the local flavor.
2. "quiet" is the wrong word to use here. It should be "ease", "carefree"
3. "it lives ... owns" is wrong interpretation. The poet is referring where do the peach flowers go? Here is a allusion to the story of the "peach blossom paradise".
4. The last line gives you the answer, to a fairyland etc.
Other than these four points, you translate nicely. In Chinese, you don't put the title first. The author's name and title appear after the poem and finally the date comes last. They are to be in a smaller sized script. If you have a seal, include it last.
This is one of my favorite poems. This translation (not my own) is superior because it rhymes in English.
Li Po's is known as the "Immortal's Poet" because most of his poems are about the fairyland, flying straight off to the moon, etc.