For a relatively recent discussion, see post 70 in this thread.
This is a quantitative questions with a mix of ludicrously big numbers and ludicrously small numbers, some in relation to space and some in relation to time, so intuition won’t be a great help. In practice, annihilation signatures should be visible – and have not been detected. However, the constraints from direct searches for antimatter streaming through our backyard (particularly light anti-nuclei) might feel more direct anyway.
This is unknown. Neutrinos definitely have two states that are to good approximation distinct, but whether these are a neutrino and antineutrino or whether these are two spin states of a single particle remains an open question.
As for the original question, as your ship approaches the possible antigalaxy, your technology would surely be good enough to notice that the cosmic rays streaming out of the various astrophysical engines of said galaxy all had the wrong polarity, and you could slam on the brakes in time.