Сестра не любит, когда паук спускается с потолка, и сразу убирает паутину.

Russian usually does not use a dummy object like English it in this pattern.

So:

• Сестра не любит, когда... = literally Sister doesn’t like when...
• natural English = Sister doesn’t like it when...

This is very common in Russian:

• Я люблю, когда идёт снег. = I love it when it snows.
• Он не любит, когда шумно. = He doesn’t like it when it’s noisy.

So nothing is missing in Russian; English just needs it here, while Russian does not.

Here когда means when / whenever, not a one-time question word.

After verbs like любить, не любить, ненавидеть, нравиться, Russian often uses:

• любить / не любить, когда...
• нравится, когда...

This means to like / not like it when...

So:

• Сестра не любит, когда паук спускается с потолка
  = Sister doesn’t like it when a spider comes down from the ceiling

If you used если, the meaning would become more conditional: if a spider comes down. That is not the normal choice in this structure.

Because Russian present tense often describes a habitual or repeated situation, just like English can.

Here the sentence is not necessarily about one spider descending right now. It describes what usually happens or what she generally reacts to:

• паук спускается = a spider comes down / descends
• сестра ... убирает паутину = the sister ... removes the web

So the present tense here has a general meaning, similar to English:

• She doesn’t like it when a spider comes down from the ceiling and immediately removes the web.

It is not limited to right now.

Because спускаться / спуститься means to go down, descend, while спускать / спустить usually means to lower something, let something down, or bring something down.

Compare:

• Паук спускается с потолка. = The spider is descending from the ceiling.
• Она спускает шторы. = She lowers the blinds.

The -ся form is used because the subject itself is moving downward.

So:

• паук спускается = the spider descends
• паук спускает would suggest the spider is lowering something else

Russian uses с + Genitive for movement from a surface or from above/off of something.

So:

• с потолка = from the ceiling / off the ceiling

That fits a spider descending from the ceiling.

By contrast:

• из + Genitive means out of the inside of something

So из потолка would sound like the spider is coming out from inside the ceiling, which is a different image.

Because the preposition с here requires the genitive case when it means from/off.

Base form:

• потолок = ceiling

Genitive singular:

• потолка

So:

• с потолка = from the ceiling

This is a very common pattern:

• с крыши = from the roof
• со стола = from the table
• с дерева = from the tree

Because паутина is the direct object of убирает, so it takes the accusative case.

Base form:

• паутина = web / cobweb

Accusative singular:

• паутину

So:

• убирает паутину = removes the web / cobweb

This is the normal feminine -a → -у pattern in the singular accusative:

• книга → книгу
• машина → машину
• паутина → паутину

It is the sister.

The structure is:

• main clause: Сестра не любит ... и сразу убирает паутину
• subordinate clause: когда паук спускается с потолка

So убирает goes with Сестра, not with паук.

In other words:

• Сестра = subject of не любит and убирает
• паук = subject only inside the когда clause

Russian often omits the subject in the second verb if it is the same as before.

So the full sense is:

• The sister doesn’t like it when a spider comes down from the ceiling, and she immediately removes the web.

Because that part is a subordinate clause introduced by когда.

Russian punctuation normally puts a comma before a subordinate clause:

• не любит, когда...

And here the subordinate clause sits between two parts of the main clause, so it is also closed with a comma:

• Сестра не любит, [когда паук спускается с потолка], и сразу убирает паутину.

So the commas show the boundaries of the когда clause.

It can mean both, depending on context.

With a spider in the sentence, паутина is naturally understood as spider web / cobweb.

So in this sentence, English could translate it as:

• web
• spider web
• cobweb

The exact choice depends on style and context. If the idea is cleaning it away in a house, cobweb often sounds especially natural in English.