Рядом с аллеей мы увидели муравейник, а на дереве — старую паутину.

Because рядом с means next to / near, and it requires the instrumental case.

So:

• аллея = dictionary form
• с аллеей = instrumental singular

This is a very common pattern:

• рядом с домом = next to the house
• рядом с дорогой = next to the road
• рядом с аллеей = next to the alley/path

Also note that рядом by itself can mean nearby, but if you say what something is next to, you normally use с + instrumental.

Because this sentence describes location, not motion.

In Russian:

• на + prepositional = on / at a place
• на + accusative = onto a place, with motion toward it

So:

• на дереве = on the tree
• на дерево = onto the tree

Compare:

• На дереве сидела птица. = A bird was sitting in/on the tree.
• Птица села на дерево. = The bird landed onto the tree.

Here, the cobweb is already located there, so Russian uses на дереве.

Because муравейник is a masculine inanimate noun, and in the singular its accusative form is the same as its nominative form.

The verb увидели takes a direct object in the accusative case, so муравейник is accusative here. It just happens to look the same:

• nominative: муравейник
• accusative: муравейник

This is normal for many masculine inanimate nouns:

• я вижу стол
• мы увидели дом
• мы увидели муравейник

Because паутина is the direct object of увидели, so it must be in the accusative case. Since паутина is a feminine singular noun, both the noun and adjective change form:

• nominative: старая паутина
• accusative: старую паутину

So:

• старая паутина = an old cobweb/web, as the subject or dictionary-style phrase
• увидели старую паутину = saw an old cobweb/web

This is a very important pattern for feminine nouns ending in -а / -я.

The dash shows that something is left out because it is understood from the first part. Here, the omitted verb is увидели.

So the full version would be:

• Рядом с аллеей мы увидели муравейник, а на дереве увидели старую паутину.

Russian often leaves out repeated words when they are obvious. The dash helps mark that omission clearly and makes the sentence more natural and stylistically smoother.

So after the dash, you should mentally understand:

• а на дереве — [мы увидели] старую паутину

Here а links two pieces of information while also setting them side by side as a mild contrast or comparison:

• near the alley — an anthill
• on the tree — an old cobweb

А often means something like:

• and meanwhile
• whereas
• while
• as for the other thing

It does not always mean a strong contrast. Often it just organizes the sentence by presenting one item, then another in a different place or situation.

If you used и, it would sound more like simple addition. А feels more natural here because the sentence contrasts two locations and two things seen there.

Russian word order is flexible, and speakers often move elements to the beginning for topic, focus, or scene-setting.

Starting with Рядом с аллеей does this:

• it sets the scene first
• it tells the listener where to look
• it makes the sentence feel more descriptive

Both versions are possible:

• Рядом с аллеей мы увидели муравейник...
• Мы увидели муравейник рядом с аллеей...

The first one is a bit more literary or descriptive, but not unnatural.

Yes. You could repeat the verb:

• Рядом с аллеей мы увидели муравейник, а на дереве увидели старую паутину.

This is grammatically correct, but it is a little heavier because увидели is repeated. The version with the dash is more elegant and more typical in written Russian when the missing word is obvious.

You could also say:

• ...а на дереве была старая паутина.

But that slightly changes the structure and focus. It now says there was an old cobweb on the tree, instead of keeping the parallel idea we saw X there, and Y there.

Увидели is:

• past tense
• plural
• perfective
• from the verb увидеть

It agrees with the subject мы, so it takes the plural past form:

• я увидел / увидела
• ты увидел / увидела
• мы увидели

The perfective verb увидеть emphasizes the completed act of noticing/seeing something. In English, this often simply translates as saw.

Yes, около аллеи could work. Both can mean near.

But there is a slight nuance:

• рядом с = right next to, beside, close by
• около = near, in the vicinity of

So:

• рядом с аллеей suggests something fairly close to the alley
• около аллеи can be a little broader or less exact

Also, the case is different:

• рядом с аллеей → instrumental
• около аллеи → genitive

Both are common, but рядом с often feels a bit more concrete and visual.

In Russian, на дереве is the normal expression for something located on a tree, including things that English might sometimes describe as in the tree.

For example:

• птица на дереве
• яблоки на дереве
• паутина на дереве

English often says in the tree for birds because they are among the branches, but Russian commonly uses на дереве in these cases. So this is a place where you should follow Russian usage rather than translate word-for-word.

It can mean either spider web or cobweb, depending on context.

In this sentence, старую паутину is naturally understood as an old cobweb or old spider web. The adjective старую makes it sound like an old, abandoned web, which is why cobweb is often the most natural English choice.

So the word itself is паутина, and the exact English translation depends on the situation.