W łazience obok umywalki leżą pasta do zębów i szczoteczka do zębów.

Because w meaning in takes the locative case when it describes location.

• dictionary form: łazienka = bathroom
• after w for location: w łazience = in the bathroom

So łazience is the locative singular form of łazienka.

Because obok meaning next to / beside requires the genitive case.

• dictionary form: umywalka = sink
• after obok: umywalki

So obok umywalki literally means next to the sink.

Because the subject is actually two things:

• pasta do zębów
• szczoteczka do zębów

They are joined by i = and, so together they form a compound subject, and the verb must be plural.

That is why Polish uses leżą = are lying / lie, not singular leży.

Polish word order is much more flexible than English word order.

This sentence starts with the place information:

• W łazience = in the bathroom
• obok umywalki = next to the sink

Then it gives the verb:

• leżą = lie / are lying

Then it names the things:

• pasta do zębów i szczoteczka do zębów

This order sounds natural in Polish because it first sets the scene and then says what is there. English usually prefers The toothpaste and toothbrush are lying..., but Polish does not need to follow that pattern.

Because leżeć means to lie or to be lying, and Polish often uses it for objects resting on a surface.

So leżą suggests that the toothpaste and toothbrush are physically lying there, probably on or near the sink.

If you used są, it would just mean are there / are present, which is more neutral.
Leżą is more vivid and specific.

Here do + genitive expresses purpose: something is for something.

So:

• pasta do zębów = toothpaste, literally paste for teeth
• szczoteczka do zębów = toothbrush, literally little brush for teeth

This is a very common Polish pattern:

• kubek do kawy = cup for coffee
• szczotka do włosów = hairbrush, literally brush for hair

Zębów is the genitive plural form of zęby = teeth.

It is genitive because do requires the genitive.
It is plural because these expressions are normally built with teeth, not tooth, in Polish.

So even though English says toothpaste and toothbrush, Polish says:

• pasta do zębów = paste for teeth
• szczoteczka do zębów = brush for teeth

That is just the standard idiomatic form.

It looks similar, but here it does not mean Italian pasta.

In Polish, pasta can mean a paste-like substance, including:

• pasta do zębów = toothpaste
• pasta do butów = shoe polish

If you want pasta as food, Polish usually says makaron.

So this is a classic vocabulary trap for English speakers.

Szczoteczka is a diminutive form of szczotka.

• szczotka = brush
• szczoteczka = little brush

A toothbrush is normally called szczoteczka do zębów, not szczotka do zębów, because it is a small personal brush. Polish uses diminutives very often in everyday vocabulary, and in many cases they are just the normal word for a smaller object.

No. Polish does not have articles like English a / an / the.

So W łazience obok umywalki leżą pasta do zębów i szczoteczka do zębów can mean:

• In the bathroom, next to the sink, there is toothpaste and a toothbrush
• or ... the toothpaste and the toothbrush
• or simply ... toothpaste and a toothbrush

The exact meaning depends on context.

That is a very good question, because umywalki could match more than one form in isolation.

For umywalka, umywalki can be:

• genitive singular
• nominative plural
• accusative plural

But here the preposition obok tells you what it must be. Since obok requires the genitive, umywalki here must be genitive singular.

So grammar around the word often tells you which form it is.

Yes. Polish does not need a separate word like English there in sentences like this.

English often says:

• There are toothpaste and a toothbrush next to the sink

Polish can simply say:

• obok umywalki leżą pasta do zębów i szczoteczka do zębów

The idea of there are is understood from the structure and context.