Plot hole: When Spock analyzes the call of the probe and compares it in a phylum search, he's presumably interfacing with the Federation memory bank as mentioned in Sulu's pre-launch status. However, the alien probe was disrupting all power and communications in the vicinity of Earth. It's extremely unlikely the ship could've interfaced with the Federation memory bank at the time of that search. (00:24:55)
Plot hole: If the Bird-of-Prey went to warp speed right after the whales were beamed up the ship would have already been so far away from Earth it would have been on its way back towards the sun, yet minutes later the ship is shown leaving the Earth, still at warp speed.