Hey there!
For the first one, they all make sense. When it comes to using them, it depends on the scenario. If you are talking about acorn woodpeckers in general, with no context, then "gathers" is the correct answer. If you are talking about in the past, "gathered" is correct. If you are talking about future tense, it is "will gather," and "is gathering" is present tense.
For the second one, the only correct answer is "birds". "bird's" is incorrect, because the "'s" is a contraction for "bird is", and "The bird is gather acorns in a granary" doesn't make sense. "birds'" doesn't make sense either, because you would use the apostrophe at the ends when talking about more than one bird in possession of something. For example, "Those are the birds' acorns" would represent more than one bird owning acorns. "birds's" is completely grammarly incorrect, and would never be used.
For the third one, any of them can be correct except the third one. "is storing" represents singular present tense, and this is plural, so it's incorrect. "are storing" would be correct, and would represent present tense. "stored" is past tense, and "will store" is future tense.
Have a terrificly amazing day! :)