После дождя скамейка была сырой.

Because после requires the genitive case.

• Dictionary form: дождь = rain
• After после: после дождя = after the rain / after rain

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после дождя = after the rain

So the form дождя is the genitive singular of дождь.

Because скамейка is a feminine singular noun, and Russian past-tense verbs agree with the subject in gender and number.

• скамейка = bench, feminine
• была = was, feminine singular

Compare:

• стол был = the table was
• скамейка была = the bench was
• окно было = the window was
• скамейки были = the benches were

So была matches скамейка.

Here сырой is the instrumental feminine singular form of the adjective сырой, used after была.

So the structure is:

• скамейка — subject
• была — was
• сырой — damp/wet, in the instrumental

Russian often uses the instrumental case for a predicate adjective after быть in the past or future.

So:

• Скамейка была сырой. = The bench was damp.

You may also hear Скамейка была сырая. That is possible too, with the adjective in the nominative. But была сырой is a very standard pattern.

Also, сырой here is not a short adjective.
The short feminine form would be сыра, which is different.

In this sentence, сырой means damp, wet, or moist.

This word can also mean raw in other contexts:

• сырое мясо = raw meat
• сырая скамейка / скамейка была сырой = a damp bench / the bench was damp

So the meaning depends on context.

For a bench after rain, сырой is understood as damp/wet, not raw.

Both can relate to wetness, but the nuance is a little different.

• мокрый = wet, soaked, clearly wet
• сырой = damp, moist, not fully dry

So:

• мокрая одежда = wet clothes
• сырая скамейка = a damp bench

After rain, a bench is often described as сырая or сырой because it may be damp rather than dripping wet. But мокрой could also work if you want to emphasize that it was really wet.

Because Russian has no articles like a or the.

So скамейка can mean:

• a bench
• the bench

And после дождя can mean:

• after rain
• after the rain

Context tells you which meaning is intended. In this sentence, English will usually translate it as the bench and the rain, but Russian does not need separate words for that.

Yes. Russian word order is flexible, although some orders sound more natural in certain contexts.

The given sentence:

• После дождя скамейка была сырой.

puts после дождя first, which sets the scene: after the rain.

You could also say:

• Скамейка была сырой после дождя.

This means essentially the same thing, but the focus is slightly different. Starting with после дождя is very natural when you want to introduce the time or situation first.

No. In the present tense, Russian usually does not use a word for am / is / are.

So:

• Скамейка сырая. = The bench is damp.
• После дождя скамейка сырая. = After the rain, the bench is damp.

But in the past tense, Russian does use был / была / было / были:

• Скамейка была сырой. = The bench was damp.

So the sentence is a good example of how Russian handles to be differently in the present and past.