Moja szczoteczka do zębów jest obok kubka.

Because moja has to agree with szczoteczka, and szczoteczka is a feminine singular noun.

In Polish, possessive words like my change form to match the noun they describe:

• mój for masculine singular
• moja for feminine singular
• moje for neuter singular

So:

• moja szczoteczka = my toothbrush
• but mój kubek = my cup/mug

This is a very common agreement pattern in Polish.

Szczoteczka is the normal word for a small brush, and it is the standard word used in szczoteczka do zębów = toothbrush.

It comes from szczotka = brush, but szczoteczka is a smaller/diminutive form. In many cases, that diminutive form became the ordinary everyday word for certain objects.

So even though it may look like little brush, in practice szczoteczka do zębów is just the normal way to say toothbrush.

This is a very common Polish pattern:

• noun + do + genitive

It often describes what something is for.

So:

• szczoteczka do zębów = literally something like brush for teeth
• natural English translation: toothbrush

Even though do often means to in other contexts, here it is part of a fixed and very common structure meaning for or used for.

You will see the same pattern in many everyday expressions, for example:

• pasta do zębów = toothpaste
• maszynka do golenia = razor / shaving machine
• okulary do czytania = reading glasses

Because the preposition do requires the genitive case, and the genitive plural of zęby is zębów.

The forms are:

• ząb = a tooth
• zęby = teeth
• zębów = of teeth / for teeth, depending on context

So in szczoteczka do zębów, the phrase must use the genitive:

• do zębów, not do zęby

Also, Polish normally uses the plural here, because a toothbrush is understood as something for brushing teeth in general, not just one tooth.

Because obok normally requires the genitive case, so kubek changes to kubka.

The basic form is:

• kubek = cup / mug

After obok:

• obok kubka = beside the cup/mug

This is similar to English prepositions forcing certain structures, except in Polish they often force a specific case.

There is also a small stem change here: the e in kubek disappears in some forms:

• kubek
• kubka
• kubkiem

That kind of vowel drop is very common in Polish nouns.

In normal use, yes: obok is followed by the genitive.

Examples:

• obok domu = next to the house
• obok stołu = next to the table
• obok kubka = next to the cup/mug

With pronouns:

• obok mnie = next to me
• obok ciebie = next to you
• obok niego = next to him

So if you learn obok + genitive, you will be on solid ground.

Usually no, not in this kind of sentence.

Here jest means is, and it connects the subject with the location:

• Moja szczoteczka do zębów jest obok kubka.

Without jest, the sentence would sound incomplete or unnatural in standard Polish.

Polish sometimes omits to be in certain very informal or special structures, but in a basic location sentence like this, jest should stay.

You could also use a more specific verb instead of jest, for example:

• Moja szczoteczka do zębów leży obok kubka. = My toothbrush is lying next to the cup.
• Moja szczoteczka do zębów stoi obok kubka. = My toothbrush is standing next to the cup.

But if you want the neutral sentence, jest is exactly right.

No, Polish word order is fairly flexible.

The sentence:

• Moja szczoteczka do zębów jest obok kubka.

is a neutral, natural order.

But other orders are possible, depending on what you want to emphasize:

• Obok kubka jest moja szczoteczka do zębów.
  Focuses more on the location.
• Moja szczoteczka jest obok kubka.
  Shorter if the context already makes do zębów obvious.

Even though Polish allows flexibility, the version you were given is a very good default pattern for learners.

It can mean either, depending on context.

Kubek is a broad everyday word for a handled drinking vessel, often something like a mug, but in some contexts English may translate it as cup.

So in this sentence, kubka could be understood as:

• beside the cup
• beside the mug

The best translation depends on the actual object being described.

A rough English-style approximation is:

MO-ya shcho-TETCH-ka do ZEM-boof yest O-bok KOOP-ka

A few helpful notes:

• moja sounds roughly like MO-ya
• szcz is a difficult Polish cluster; in approximation, shch is close enough for a learner
• zębów is pronounced roughly ZEM-boof in normal speech
• jest sounds like yest
• kubka is pronounced roughly KOOP-ka because the b is devoiced before k

Stress in Polish is usually on the second-to-last syllable, so:

• MO-ja
• szczo-TECZ-ka
• ZĘ-bów
• O-bok
• KUB-ka

You do not need perfect pronunciation immediately, but it helps to notice these stress and sound patterns early.