Why is ammonia a stronger base than water and, following from that, why is phenylamine basic while phenol is acidic?

I don't know. Why?

Water has oxygen and is electronegative and has no electrons to share, while ammonia has nitrogen which has two free electrons (1 electron pair) available to bind. (Draw a lewy dot structure, it'll make more sense.) Thus making the ammonia more basic.

If you want to delve further into it, it's also related to orbitals.

Ammonia has one unshared pair of electrons, water has two. In our part of the universe, the more unshared pairs you have, the stronger the acid you are (Look at the next one - HF)

The phenyl derivatives of NH3 and H20 have lower electron density because the benzene ring is electron withdrawing. Thus Phenol is truly a weak acid and Aniline is a weaker base than their parent compounds.