Распылитель лежит рядом с аквариумом.

Because рядом с requires the instrumental case.

• dictionary form: аквариум
• instrumental singular: аквариумом

So:

• рядом с аквариумом = next to the aquarium

This is a very common pattern:

• рядом с домом = next to the house
• рядом с окном = next to the window
• рядом с машиной = next to the car

Here с is part of the fixed expression рядом с, which means next to / beside.

So you should learn it as one unit:

• рядом с + instrumental

Even though с often means with, in this expression it does not mean with in the usual sense. It is just part of how Russian says next to.

Лежит already includes the idea of is.

It comes from the verb лежать, which means to lie or to be lying. So:

• лежит = is lying

Russian normally does not use a separate present-tense verb to be the way English does. So instead of something like is lying, Russian just says лежит.

In Russian, verbs like лежать, стоять, and висеть are used very naturally for objects too.

Russian often describes an object’s position more specifically than English does:

• лежать = to be lying, resting horizontally, or just placed somewhere
• стоять = to be standing upright
• висеть = to be hanging

So if the sprayer is resting on a surface rather than standing upright, лежит is a natural choice.

Sometimes yes, but it changes the image.

• лежит suggests the object is lying/resting on something
• стоит suggests it is upright/standing

For many small objects, Russian chooses between these based on how the speaker imagines the object positioned. If the thing is on its side or simply placed there, лежит is more likely.

Because it is the subject of the sentence.

The subject of a normal Russian sentence is usually in the nominative case, which is the dictionary form. Here:

• распылитель = the thing doing the action/state
• лежит = is lying
• рядом с аквариумом = next to the aquarium

So распылитель stays in the nominative.

The dictionary form is лежать.

Лежит is the 3rd person singular present tense form:

• я лежу = I am lying
• ты лежишь = you are lying
• он / она / оно лежит = he / she / it is lying
• они лежат = they are lying

In this sentence, the subject is singular, so we use лежит.

Not by itself, not exactly. It is often treated as an adverb, but in this meaning it commonly appears in the pattern:

• рядом с + instrumental

So for a learner, the most useful approach is to memorize the whole construction рядом с rather than worrying too much about the label.

Yes. Russian word order is more flexible than English word order.

These are all possible, with slightly different emphasis:

• Распылитель лежит рядом с аквариумом.
  Neutral statement.

• Рядом с аквариумом лежит распылитель.
  More emphasis on the location: next to the aquarium.

The meaning stays basically the same because the case endings show the grammatical roles.

All of them can often be translated as near / by / next to, but they are not always identical.

• рядом с = next to, right beside
• возле = near, by
• около = near, around; also has other uses

In many everyday contexts they overlap:

• рядом с аквариумом
• возле аквариума
• около аквариума

But рядом с often gives the strongest sense of being immediately beside something.

Also note the case difference:

• рядом с
  • instrumental
• возле / около
  • genitive

It literally refers to something that sprays or disperses something. The exact English word depends on context.

In an aquarium context, it may refer to something like:

• a sprayer
• an atomizer
• a diffuser
• an air stone / spray device

So the Russian word is broad, and the best English translation depends on what object is actually meant.

A rough pronunciation guide:

• Распылитель ≈ ra-spi-LI-ti-elʹ
• лежит ≈ li-ZHEET
• рядом ≈ RYA-dam
• с ≈ s
• аквариумом ≈ ak-VA-ri-u-mam

A more natural flow would sound roughly like:

• ra-spi-LI-ti-elʹ li-ZHEET RYA-dam s ak-VA-ri-u-mam

The stress is on:

• распылитель
• лежит
• рядом
• аквариумом

In some contexts, yes, especially in informal speech, signs, labels, or very telegraphic style. But as a full normal sentence, лежит makes it complete and natural.

So:

• Распылитель рядом с аквариумом = possible, but more clipped
• Распылитель лежит рядом с аквариумом = full standard sentence

Russian often omits the present tense of to be, but here лежит is not just is — it adds the important idea is lying.