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It might have needed to transition to the sunny spot a little more gradually. It's probably quite a bit hotter outside than in your house, too so the plant would need more water than you've normally given it. This would explain the droopiness. I'd recommend you move it to a shady spot until it recovers, then gradually move it into more and more sun. Once it's a "real" tree it wants all sun, all day.
As long as your pot has good drainage, the extra rain should be fine for it. Avocadoes grow here and get drenched regularly through the summer.
Yes, they grow straight up for a while before they branch. Even mature trees are on the tall and narrow side from what I've seen unless they've been pruned to stay shorter. (not something you want to do to a baby).
