Learn Hardcore Russian: На новой кровати лежит тяжёлый матрас, а рядом — чистая наволочка и новый пододеяльник. - A heavy mattress is lying on the new bed, and next to it there is a clean pillowcase and a new duvet cover.

Because На новой кровати is a location phrase (meaning on the new bed), and Russian often puts location first to set the scene. The “real” grammatical subject of the first part is тяжёлый матрас (the thing that is lying there).

Why is it на новой кровати and not в новой кровати?

на + Prepositional is used for surfaces and “on top of” meanings: на кровати = on the bed.
в + Prepositional usually means “inside” something: в коробке (in a box), в комнате (in a room). A bed is treated as a surface for this meaning.

What case is новой кровати, and why that ending?

It’s Prepositional case after на (location use): на чём? → на кровати.
кровать is feminine, and in the prepositional singular it becomes кровати; the adjective agrees: новой (fem., prep. sing.).

Why is the verb лежит before the noun матрас?

This is a common Russian word order when you introduce something new in a location:

На кровати лежит матрас ≈ “There is a mattress lying on the bed.” It’s similar to English On the bed lies a mattress, which also puts the location first and the new item later.

Why is it лежит (singular) and not plural, since more things are mentioned later?

The first clause has one subject: тяжёлый матрас (singular), so the verb is singular: лежит.
The other items (наволочка и пододеяльник) belong to the second part of the sentence, where the verb is simply omitted (see the dash question below).

What does the dash — mean in а рядом — чистая наволочка и новый пододеяльник?

The dash marks ellipsis: something is intentionally left out because it’s understood from context.
Here it’s like:

а рядом (лежит/лежат/находится/находятся) чистая наволочка и новый пододеяльник In English you might do something similar with “and nearby, a clean pillowcase and a new duvet cover.”

Shouldn’t it be рядом лежат (plural), since there are two things?

If you include the verb, plural is natural: рядом лежат наволочка и пододеяльник.
But because the verb is omitted, the sentence avoids committing to a specific verb form; it just lists what is there nearby.

Why is рядом used without с here? I’ve seen рядом с.

Two different structures:

рядом (adverb) = nearby / next to it (in general)
Example: Рядом — магазин.
рядом с + Instrumental = next to (something specific)
Example: Рядом с кроватью — стол. In your sentence, рядом works because the reference point is already clear (the bed/the mattress/the scene).

What case are матрас, наволочка, пододеяльник in?

They are in the Nominative case because they are (explicitly or implicitly) the subjects / items being stated as present:

лежит (кто? что?) матрас
рядом — (кто? что?) наволочка и пододеяльник

How do the adjectives agree: тяжёлый, чистая, новый?

Adjectives agree with nouns in gender, number, and case:

тяжёлый матрас: masculine, nominative singular
чистая наволочка: feminine, nominative singular
новый пододеяльник: masculine, nominative singular
Also на новой кровати: feminine, prepositional singular.

Why is there a comma before а?

Because а here connects two parts of a compound sentence:
[On the bed lies a mattress], and [nearby — a pillowcase and a duvet cover].
Russian typically uses a comma before а in this kind of structure.

What’s the difference between наволочка and пододеяльник?

наволочка = pillowcase (cover for a pillow: подушка)
пододеяльник = duvet cover / blanket cover (cover for a duvet/blanket: одеяло)
So the sentence lists bedding items: mattress + pillowcase + duvet cover.

How is тяжёлый pronounced in this sentence, and what about ё in тяжёлый?

тяжёлый is pronounced with ё as yo: roughly tya-ZHYOL-y (stress on ё).
In Russian writing, ё is sometimes printed as е, but the pronunciation remains yo.