لم يعد هناك برغوث في فراء القطة.

In this sentence, لم يعد is a very common expression meaning no longer or not anymore.

So:

• لم يعد هناك برغوث... = There is no longer a flea...

Although the verb comes from عاد (to return), in this pattern the whole expression usually means that a previous situation has stopped being true.

Because لم is followed by the imperfect/present form of the verb, not the past.

The verb is:

• past: عاد
• imperfect: يعود

After لم, the imperfect goes into the jussive. With this kind of weak verb, the long vowel drops:

• يعود → يَعُدْ

In normal unvowelled writing, that appears as يعد.

So لم يعد is the correct form.

Good question. Grammatically, لم negates the imperfect and often gives a past sense. But لم يعد is idiomatic: it means that something used to be true, but now it is not.

So the focus is on a change of state:

• before: there was a flea in the cat's fur
• now: that is no longer true

That is why English usually translates it with no longer or not anymore, not just a simple past negation.

Here هناك works like existential there in English:

• هناك برغوث = There is a flea

So in this sentence:

• لم يعد هناك برغوث = There is no longer a flea

In other contexts, هناك can literally mean there / over there, but here it is mainly part of the there is / there are structure.

Because in Arabic, present-tense sentences often do not use a separate verb meaning is/are.

So Arabic can simply say:

• هناك برغوث = There is a flea

There is no need for a present-tense equivalent of is here. The meaning is understood from the structure itself.

Because existential sentences like this usually introduce something as indefinite:

• برغوث = a flea

That matches English:

• There is a flea
• not usually There is the flea

Using البرغوث would sound much more specific, as if both speaker and listener already know exactly which flea is being discussed.

So هناك برغوث is the natural choice.

If fully vocalized, it would normally be بُرغوثٌ with nominative ending.

So the fully vowelled phrase would be:

• لَمْ يَعُدْ هُناكَ بُرغوثٌ في فِراءِ القِطَّةِ

In traditional grammar, برغوثٌ is the noun whose existence is being stated, so it appears in the nominative. In everyday unvowelled writing, you just see برغوث without the ending.

It is an iḍāfa construction, often called a construct phrase.

Literally:

• فراء القطة = fur of the cat

Natural English:

• the cat's fur

In an iḍāfa:

1. the first noun comes first: فراء
2. the second noun specifies it: القطة

A key point: the first noun normally does not take ال in this structure. The definiteness of the whole phrase comes from the second noun.

Because القطة is definite, فراء القطة means the cat's fur, not just cat fur.

They are genitive for two different reasons:

• فراء is genitive because it comes after the preposition في
  • في فراءِ
• القطة is genitive because it is the second term of the iḍāfa
  • فراءِ القِطَّةِ

So in full form:

• في فِراءِ القِطَّةِ

Both end up genitive, but not for the same grammatical reason.

Yes. Arabic allows some flexibility in word order.

For example, you could also say:

• لم يعد في فراء القطة برغوثٌ

This still means essentially the same thing: There is no longer a flea in the cat's fur.

You may also see a version with يوجد:

• لم يعد يوجد برغوثٌ في فراءِ القطةِ

That is also correct, but لم يعد هناك... is very natural and common.

The original sentence is a straightforward, idiomatic way to say it.