В старом доме была сильная сырость, поэтому на стене появилось пятно.

Because в means in here, and after в for location, Russian uses the prepositional case.

• дом → доме = in the house
• старый → старом to match the noun in case, gender, and number

So:

• в доме = in the house
• в старом доме = in the old house

A useful contrast:

• в доме = in the house
• в дом = into the house

So в старом доме answers where?, not where to?

Russian usually leaves out to be only in the present tense.

So:

• В доме сильная сырость. = There is strong dampness in the house.
• В доме была сильная сырость. = There was strong dampness in the house.

In the past tense, Russian normally does use a form of быть:

• был masculine
• была feminine
• было neuter
• были plural

So была is necessary here because the sentence is in the past.

Because the verb agrees with сырость, and сырость is a feminine singular noun.

So the past-tense verb must also be feminine singular:

• сырость была

Even though English says there was, Russian grammar still treats сырость as the grammatical subject, so the verb matches it.

Because сильная has to agree with сырость.

Сырость is:

• feminine
• singular
• nominative

So the adjective must also be:

• feminine
• singular
• nominative

That gives сильная.

This phrase means something like:

• strong dampness
• severe dampness
• a lot of damp

Сырость means dampness, moisture in the air or walls, or a generally damp condition.

It is not the same as just water. It suggests that a place feels wet, clammy, humid, or badly ventilated.

So in this sentence, сильная сырость suggests the old house had a serious damp problem, which then caused the stain.

Поэтому means therefore, so, or that’s why.

It connects the first idea with the result:

• there was strong dampness
• therefore a stain appeared

The comma is used because the sentence has two parts:

• В старом доме была сильная сырость
• поэтому на стене появилось пятно

In English, this works a lot like so or therefore in a compound sentence.

Because на стене means on the wall as a location.

Russian uses:

• на + prepositional for location
• на + accusative for movement toward a surface

So:

• на стене = on the wall
• на стену = onto the wall

Here the sentence is telling us where the stain appeared, so Russian uses the location form:

• на стене появилось пятно = a stain appeared on the wall

Because the subject is пятно, and пятно is neuter singular.

So the past-tense verb must also be neuter singular:

• пятно появилось

The infinitive is появиться = to appear, and появилось is the past-tense neuter form.

Появилось is perfective, so it presents the event as a single completed result: the stain appeared.

That fits the meaning well, because the sentence describes a consequence of the dampness.

By contrast, появлялось would be more like:

• was appearing
• used to appear
• appeared repeatedly

So появилось пятно sounds natural for a stain appeared as one resulting event.

Because пятно is the grammatical subject of появилось.

In Russian, the thing that appears is usually in the nominative:

• появилось пятно = a stain appeared
• literally: a stain appeared

Even though English learners may focus on on the wall, that phrase is just the location. The actual subject is пятно.

No, Russian word order is fairly flexible. The version here is neutral and natural:

• В старом доме была сильная сырость, поэтому на стене появилось пятно.

But other orders are possible if you want different emphasis. For example:

• На стене появилось пятно, потому что в старом доме была сильная сырость.
• Сильная сырость была в старом доме, поэтому на стене появилось пятно.

The cases show the grammatical roles, so Russian does not depend on word order as much as English does. Still, some orders sound more natural than others, and the original sentence is a very normal one.

Russian has no articles, so it does not have separate words for a/an and the.

That means:

• в старом доме can mean in an old house or in the old house
• появилось пятно can mean a stain appeared or the stain appeared

The exact meaning depends on context. In many cases, Russian leaves this unstated unless it really matters.