Answer: Based on your description it sounds like some form of caterpillar (rather than spiders or ants) may be eating your tree and nesting in it thus causing the stringy material. Although some caterpillars are not particularly harmful to an established healthy tree, it may be a good idea to remove these from your baby tree while it tries to become established. You may be able to do it by hand if the tree is small enough or you may need to apply Bt according to the label instructions. However, you might wish to take a sample to your County Extension (305-5742) for a more specific identification of the problem and suggested controls, if any.
Also, the droopy leaves may be a cause of concern. Droopy leaves can be caused by a number of things, but some common reasons include overwatering, underwatering, and late afternoon heat stress (you may notice that the tree wilts a bit in the late afternoon on a hot sunny day but looks fine again the next morning). Make sure that you water the tree deeply once a week -- check about an inch down into the soil at the base of the trunk to see if it is moist. This is better than a daily sprinkling. Also, keep a few inches of organic mulch around but not touching the trunk. Good luck with your tree!
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