На старой паутине была вода, а рядом был маленький жук.

Because на here means on, and it is used to show location.

When на means location (on, at), the noun usually goes into the prepositional case:

• паутина → на паутине
• старая паутина → на старой паутине

So:

• на = on
• старой = old, in the feminine singular prepositional form
• паутине = spider web, in the prepositional case

Together, на старой паутине means on the old spider web.

Because the noun паутина is:

• feminine
• singular
• in the prepositional case

Russian adjectives must agree with the noun they describe in:

• gender
• number
• case

So:

• nominative: старая паутина
• prepositional after на for location: на старой паутине

This is normal adjective agreement.

In the past tense, Russian verbs agree with the subject in gender and number.

Here:

• вода is feminine, so you get была
• жук is masculine, so you get был

Compare:

• вода была / была вода = the water was / there was water
• жук был / был жук = the beetle was / there was a beetle

Past tense forms:

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

Yes. In this sentence, была вода and был маленький жук are being used in an existential sense: they mean that something was present.

Russian does not use a separate word like English there in there was. Instead, it often just uses the verb to be:

• была вода = there was water
• был маленький жук = there was a little beetle

So English uses there was, but Russian simply uses was.

Russian word order is more flexible than English word order.

Both are possible, but they can sound slightly different:

• была вода often introduces water as something present in the scene: there was water
• вода была can sound more like the water was... with more focus on water

In this sentence, была вода and был маленький жук sound natural because the speaker is describing what was there in the picture or scene.

So this word order helps create a scene-setting, descriptive feeling.

Because вода is the grammatical subject of была.

Even though English says there was water, Russian still treats water as the subject in this kind of sentence, so it usually appears in the nominative case:

• вода = nominative
• жук = nominative

That is why you see:

• была вода
• был маленький жук

not accusative forms.

Рядом means nearby, next to it, or beside it.

In this sentence, it is being used as an adverb, so it does not change form.

• рядом = nearby / beside

So:

• а рядом был маленький жук = and nearby there was a little beetle

You can also see рядом с + instrumental:

• рядом с домом = next to the house
• рядом с паутиной = next to the spider web

But in your sentence, рядом stands alone and simply means nearby.

Both а and и can be translated as and, but they are not used in exactly the same way.

• и simply joins things
• а often links two parts while lightly contrasting or shifting attention

In this sentence:

• На старой паутине была вода, а рядом был маленький жук.

The speaker first mentions the water on the web, then shifts to another detail nearby: the beetle.

So а works well here because it means something like:

• and meanwhile
• and as for nearby
• while nearby

It is not a strong contrast like but, just a gentle change of focus.

Because жук is the subject of the clause, so it is in the nominative case.

• жук = nominative singular
• маленький agrees with жук in masculine singular nominative

So:

• маленький жук = a little beetle

You would get маленького жука in other situations, for example as an object:

• Я вижу маленького жука. = I see a little beetle.

But here the beetle is not the object; it is the thing that was there.

Жук is a masculine noun meaning beetle or sometimes just bug in a broader everyday sense, depending on context.

In this sentence, маленький жук most naturally means a little beetle.

It is masculine, which is why you get:

• был
• маленький

not feminine or neuter forms.

It can mean spider web in general, and in many contexts it can also be translated as cobweb.

So старая паутина could suggest:

• an old spider web
• an old cobweb

The exact English choice depends on the scene and style of translation.

Russian has no articles like English a and the.

So:

• вода can mean water or the water
• маленький жук can mean a little beetle or the little beetle

The exact meaning depends on context.

In this sentence, English would most naturally say:

• there was water
• a little beetle was nearby

because the sentence is introducing these things into the scene.

Yes. In the present tense, Russian usually omits the verb to be.

So a present-tense version would often be:

• На старой паутине вода, а рядом маленький жук.

This sounds like There is water on the old spider web, and nearby there is a little beetle.

In the past tense, however, Russian must show the verb:

• была
• был

So your sentence needs those forms because it describes a past scene.