Без наполнителя кошка смотрит на меня и не идёт в лоток.

Because без always requires the genitive in Russian.

So:

• без = without
• наполнитель = filler / litter
• без наполнителя = without litter

This is a fixed grammar rule:

• без сахара = without sugar
• без воды = without water
• без наполнителя = without litter

Here наполнителя is the genitive singular form of наполнитель.

Literally, наполнитель means filler, filling, or filling material. In the context of pets, it commonly means cat litter.

So in this sentence, без наполнителя means something like:

• without litter
• without cat litter
• without litter material

Russian often uses a more general word where English prefers a more specific phrase.

Кошка is the subject of the sentence, so it is in the nominative case.

The cat is the one doing the actions:

• смотрит = looks
• не идёт = does not go

So кошка stays in the basic dictionary form:

• кошка = cat

If the cat were the object, then you might see кошку instead.

Because the verb смотреть normally uses the preposition на when it means to look at someone or something.

So:

• смотреть на кого? что? = to look at someone/something

That is why you get:

• смотрит на меня = looks at me

And because на here expresses the target of looking, меня is in the accusative case.

Compare:

• Я смотрю на кошку. = I look at the cat.
• Кошка смотрит на меня. = The cat looks at me.

Меня is the accusative (and also genitive) form of я.

After смотреть на, Russian uses the accusative:

• я = I
• меня = me

So:

• на меня = at me

This is just like English changing I to me.

Идёт is the 3rd person singular form of идти.

• идти = to go, to be going, to move in one direction
• идёт = he/she/it goes, is going

Here the subject is кошка, so:

• кошка идёт = the cat goes / is going

In context, не идёт в лоток means the cat does not go into the litter box, or more naturally, the cat will not use the litter box.

This is a very common question, because Russian has two basic verbs for to go on foot:

• идти = to go in one direction / be on the way
• ходить = to go habitually, repeatedly, or in different directions

In real usage, идти в лоток is common when talking about whether an animal goes into the litter box at a given moment or in a concrete situation.

So не идёт в лоток can sound like:

• is not going into the litter box
• won’t go into the litter box

If you said не ходит в лоток, that would more strongly suggest a habit:

• does not use the litter box in general
• doesn’t go to the litter box as a regular behavior

So the choice affects the nuance.

Because в can take either the accusative or the prepositional, depending on meaning.

• в + accusative = into, motion toward the inside of something
• в + prepositional = in, inside, location

Here the idea is motion:

• идёт в лоток = goes into the litter box

So Russian uses the accusative:

• лоток → в лоток

Compare:

• Кошка в лотке. = The cat is in the litter box.
  
  • location, so prepositional
• Кошка идёт в лоток. = The cat goes into the litter box.
  
  • motion, so accusative

Лоток literally means tray or box-like tray. In pet contexts, it commonly means a litter tray or litter box.

So:

• кошачий лоток = cat litter box
• идти в лоток = to go to the litter box / use the litter box

Russian often leaves out words that are obvious from context. Since the sentence already mentions a cat and litter, лоток is naturally understood as the cat’s litter box.

Because Russian often omits a repeated subject when it is already clear.

So:

• Без наполнителя кошка смотрит на меня и не идёт в лоток.

means:

• Without litter, the cat looks at me and does not go into the litter box.

The subject кошка applies to both verbs:

• смотрит
• не идёт

English does the same thing:

• The cat looks at me and does not go into the litter box.

You would not normally repeat the cat there either.

Yes, Russian word order is fairly flexible.

This sentence begins with Без наполнителя, which sets the condition first:

• Без наполнителя = without litter

Then comes the main subject and actions:

• кошка смотрит на меня и не идёт в лоток

This order is natural because it foregrounds the cause or circumstance first.

Other word orders are possible, for example:

• Кошка без наполнителя смотрит на меня и не идёт в лоток.
• Кошка смотрит на меня и не идёт в лоток без наполнителя.

But these may sound less natural or may slightly shift what is being emphasized.

The original sentence is a very normal way to present the idea.

It can suggest either, depending on context.

Literally:

• не идёт = does not go / is not going

But when talking about animals and behavior, it often has the practical sense of:

• won’t go
• refuses to go
• doesn’t use it

So in this sentence, the meaning is probably not just physical movement, but behavior: the cat will not use the litter box if there is no litter in it.

Yes, a couple of useful points:

• наполнителя → stress on ни: на-пол-ни-те-ля? Wait, actually the common stress is наполни́теля
• смотрит → stress on the first syllable: смо́трит
• меня → stress on the second syllable: ме-ня́
• идёт → stress on ё: и-дёт
• лоток → stress on the second syllable: ло-то́к

Also, ё in Russian is sometimes written as е in ordinary texts, so you may see идет, but it is still pronounced идёт.