Баррель лежит рядом с хвостовиком.

Because рядом с requires the instrumental case.

• dictionary form: хвостовик
• instrumental singular: хвостовиком

So in this sentence, с хвостовиком means next to the shank/stem/tailpiece.

This is a very common Russian pattern: a preposition determines the case of the noun that follows it.

Because рядом с is the normal expression for next to / beside.

Compare:

• Он рядом. = He is nearby.
• Он рядом с домом. = He is next to the house.

So:

• рядом by itself = nearby
• рядом с + instrumental = next to something

Russian often uses a position verb where English would simply use is.

Here, лежит comes from лежать, meaning to lie. It suggests that the object is in a lying or resting position.

Russian commonly distinguishes:

• лежать = to be lying
• стоять = to be standing
• висеть = to be hanging

So Баррель лежит... sounds natural if the barrel is on its side or simply resting somewhere. A more neutral option would be находится, but that is more formal and less visual.

Usually, yes, or at least that is the basic image.

For objects, Russian often chooses the verb based on how the object is positioned:

• Книга лежит на столе. = The book is lying on the table.
• Бутылка стоит на столе. = The bottle is standing on the table.
• Картина висит на стене. = The picture is hanging on the wall.

So if the barrel were upright, Russian speakers might prefer стоит instead of лежит.

It is in the nominative singular, because it is the subject of the sentence.

The sentence structure is:

• Баррель = subject
• лежит = verb
• рядом с хвостовиком = location phrase

Also, баррель is a masculine noun, which is typical for nouns ending in a consonant.

Because Russian has no articles.

So Баррель лежит рядом с хвостовиком can mean:

• A barrel is lying next to the tailpiece
• The barrel is lying next to the tailpiece

Which one is meant depends on the context.

If Russian wants to be more specific, it can use other tools, such as word order, context, or words like этот (this) and тот (that).

Yes. Russian word order is fairly flexible.

The given sentence:

• Баррель лежит рядом с хвостовиком.

is a neutral way to say it.

You could also say:

• Рядом с хвостовиком лежит баррель.

That still means the same thing, but it puts more focus on the location first, a bit like Next to the tailpiece lies a barrel.

So the meaning stays similar, but the emphasis changes.

Yes, you often can.

Possible alternatives:

• рядом с хвостовиком
• около хвостовика
• возле хвостовика

But notice the grammar change:

• рядом с
  • instrumental
• около
  • genitive
• возле
  • genitive

So:

• рядом с хвостовиком
• около хвостовика
• возле хвостовика

In meaning:

• рядом с often feels like right next to / beside
• около and возле can be slightly broader, more like near / by

In many real contexts, though, they are very close in meaning.

Yes. Russian does not need a separate word for it here.

English says:

• It is lying next to...

Russian usually just names the thing itself:

• Баррель лежит рядом с хвостовиком.

That is fully natural. Since баррель is already the subject, there is no need for an extra pronoun.