Hay una gotera en el techo y el agua cae cerca de la tubería.

Hay is the impersonal form of haber meaning there is / there are. It doesn’t change for singular vs plural:

• Hay una gotera = There is a leak/drip.
• Hay dos goteras = There are two leaks/drips.

It’s not the same as estar or ser. You generally use hay to introduce the existence/presence of something.

In Spain:

• techo = the ceiling (the inside surface you see from inside a room).
• tejado = the roof (the outside roof structure).

So Hay una gotera en el techo usually means the water is dripping from the ceiling inside, even if the underlying cause is the roof.

en el techo means the leak/drip is located on/in the ceiling (as a place where you observe it).
del techo would mean from the ceiling (origin), and it’s less natural for describing the existence of a leak spot. If you want to emphasize origin, you might say:

• El agua cae del techo = The water falls/drips from the ceiling.

y simply links two related facts: 1) There is a leak in the ceiling. 2) The water falls near the pipe.

It can suggest cause-and-effect contextually (the leak causes the water to fall), but grammatically it’s just and.

Literally, cae = falls (from caer). With water from a leak, Spanish commonly uses caer to describe dripping/falling water:

• El agua cae can be understood as The water is dripping/falling.

If you want to emphasize dripping as repeated drops, you might also hear:

• El agua gotea = The water drips.

Spanish often uses the present simple for actions happening right now, especially when describing a situation:

• El agua cae = The water is falling/dripping (now, as a current situation).

Está cayendo is also correct, but it can feel more “right this second / in progress”:

• El agua está cayendo = The water is falling right now.

Agua is feminine, but in the singular it takes el (not la) because it starts with a stressed a- sound, to avoid la a...:

• el agua, el águila, el arma (singular)

But adjectives and pronouns still treat it as feminine:

• el agua fría (cold water — fría feminine)
• mucha agua (a lot of water — mucha feminine)

Plural goes back to las:

• las aguas.

In standard Spanish (Spain), the common pattern is:

• cerca de + noun/pronoun = near (to) something
  So cerca de la tubería = near the pipe.

You can say cerca on its own only if the context already makes the reference clear:

• Cae cerca = It falls nearby.

• tubería = piping / pipework / a pipe system (often the installed plumbing line)
• tubo = a tube/pipe as an individual object (a piece of pipe)

In a house context, tubería often implies plumbing.

• tubo: possible if you mean a specific visible pipe, but it can sound less “plumbing system” and more “a tube.”
• cañería: understood, but it’s more common in many Latin American varieties; in Spain you’ll more often hear tubería or las tuberías (plumbing).

So for Spain, tubería is a safe, natural choice.

la tubería suggests a specific pipe that’s known in the context (e.g., the pipe you can see there, or the relevant plumbing line). If it were just “some pipe, not identified,” you could say:

• ...cerca de una tubería = near a (certain) pipe.

Both are grammatical; the definite article is often used when the object is identifiable in the scene.

Yes, word order is flexible. Starting with Hay... is common because it introduces the situation. But you can reorder for emphasis:

• El agua cae cerca de la tubería y hay una gotera en el techo.

It still means the same, but it changes what feels like the “main point.”

They’re related but different:

• gota = a drop (one unit): una gota de agua
• gotera = a leak/drip spot (the problem/phenomenon in a ceiling/roof)
• goteo = dripping as an action/process (like “dripping” in general): Hay un goteo constante (There’s constant dripping.)

In this sentence, gotera fits best for a household ceiling leak.