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Farmer's Fence: The mathematician builds a tiny fence around himself and declares himself outside, thus enclosing (almost) the entire surface of Earth.
Dog Crossing: The river is frozen.
Not Twins: They're two of three triplets.
Triangles: 35. (You might have guessed this from the hint, and from the fact that the five-fold symmetry of the figure means that any answer must be a multiple of five.)
Manhole: A square cover could fall into the hole. (Round is also stronger, weight for weight.)
Impossible: Cut yourself an inch-wide strip from a letter-size sheet of paper and play — it's easy to figure out.
Tennis Tournament: 124, the number of players who had to be eliminated.
Six Letters: BANANA
Tempus Fugit: Yes, it is!
Noun: (Declarative) Time goes by as fast as an arrow flies. Adjective: (Declarative)
Drosophila temporis, the time fly, enjoys hanging out on arrows. Verb: (Imperative) Measure the speed of flies the same way you'd measure an arrow's speed. Verb: (Imperative) Time flies, using the method arrows use to time them (Analogy: "Bend it like Beckham.") Verb: (Imperative) Measure the speed of flies that resemble arrows. (Analogy: "Eat vegetables like kale.")