tap, spigot or faucet?
Most common is faucet but tap fairly common and spigot mostly for water source outside like for watering the garden or washing the car.

the large thing in the room with all the plumbing fixures?
Toilet, sink and bathtub with shower called a bathroom. If just a basin and toilet then it's a half-bath or powder room especially if on ground floor for use by guests.

Lav to me means a lavatory which is a holdover from my convent school days ... more commonly called a Ladies Room ... toilets in cubicles and a bank of wash basins.

firemen attach their hoses to a hydrant for municipal water

protesters/rioters hauled away in Black Maria, Paddy Wagon.

what is under your house My house is slab-built which means I have no basement (old fashioned word) or cellar (more common) and tho I sometimes miss the storage a cellar offers, other times I'm glad I don't have the temptation to amass yet more "stuff!"

the water for household Hot water heater with "Quick return" ... which means my guests and I can take showers one following quickly upon another, without my having to bath in frigid aqua. Heavenly!