На уроке рисования у неё на столе лежали палитра и гуашь.

На уроке рисования means in/at the drawing class or more literally at the lesson of drawing.

• уроке is in the prepositional case because it follows на in the meaning of location: на уроке = in class / during the lesson
• рисования is the genitive singular of рисование (drawing, painting, art activity)

This is a very common Russian pattern:

• урок математики = a math lesson
• урок музыки = a music lesson
• урок рисования = a drawing lesson

So рисования is there because Russian often uses the genitive after a noun to show what kind of lesson something is.

Because after на when talking about location or being somewhere, Russian usually uses the prepositional case.

So:

• на уроке = in class / at the lesson
• на столе = on the table

Compare that with motion:

• на урок = to the lesson
• на стол = onto the table

So in your sentence nothing is moving to the lesson; the scene is simply taking place there. That is why it is на уроке.

In this sentence, у неё literally means by her / at her, but very naturally it often corresponds to English her in expressions like on her desk.

So:

• у неё на столе = on her desk / on the table in front of her

Russian often uses у + genitive to show possession or something associated with a person:

• У неё есть книга. = She has a book.
• У неё на столе лежали... = On her desk there lay...

So yes, it is related to the Russian way of expressing possession, but here it is part of the phrase describing location.

Because у requires the genitive case.

The pronoun она changes like this:

• nominative: она = she
• genitive: её / неё

After many prepositions, Russian adds an initial н- to pronouns:

• у неё
• к нему
• с ней
• для них

So у неё is the correct form after у.

Because на столе describes location, not direction.

• на столе = on the table (already there)
• на стол = onto the table (movement toward it)

In your sentence, the palette and gouache were already there, so Russian uses the prepositional case:

• столе = prepositional singular of стол

Because together they form a compound subject:

• палитра и гуашь = a palette and gouache

Two singular nouns joined by и usually take a plural verb:

• Книга и тетрадь лежали на столе.
• Палитра и гуашь лежали на столе.

So лежали is plural because there are two things.

Russian often prefers a position verb where English might simply say was/were.

Here лежать means to lie or to be lying. It suggests the objects were lying on the table, probably flat or simply placed there.

Compare:

• На столе были палитра и гуашь. = There were a palette and gouache on the table.
• На столе лежали палитра и гуашь. = A palette and gouache were lying on the table.

The second version is more vivid and more natural when talking about objects resting on a surface.

Russian often chooses among verbs like лежать, стоять, and висеть depending on the position of the object.

• лежать = lie, be lying
• стоять = stand, be standing
• висеть = hang, be hanging

A palette and gouache are naturally imagined as things placed on the desk, so лежали makes sense.

Sometimes with containers like a bottle or jar, Russian might use стоять if they are upright:

• На столе стояла банка гуаши. = A jar of gouache was standing on the table.

But in your sentence гуашь means the paint material, not necessarily a particular upright container, so лежали works fine with the whole phrase.

Гуашь means gouache, the type of paint. In Russian it is usually treated as a singular noun referring to the material.

So:

• гуашь = gouache paint
• купить гуашь = to buy gouache
• рисовать гуашью = to paint with gouache

It is not normally used like one gouache, two gouaches. If you want individual items, Russian would usually specify them:

• баночка гуаши = a jar of gouache
• краски гуашь = gouache paints

Also, гуашь is a feminine noun, even though it ends in -ь.

Nouns ending in -ь can be either masculine or feminine, so you often have to learn their gender individually.

Гуашь is feminine.

One clue is how it behaves in other forms:

• nominative: гуашь
• instrumental: гуашью

That -ью ending is typical of many feminine soft-sign nouns.

Other feminine nouns of this type include:

• ночь = night
• дверь = door
• тетрадь = notebook

So it is best to memorize гуашь as a feminine noun.

Russian word order is much freer than English word order because case endings show grammatical roles.

This sentence starts with the setting and location, then gives the things:

• На уроке рисования = setting
• у неё на столе = more precise location
• лежали = verb
• палитра и гуашь = subject

A very literal structure would be something like:

At the drawing lesson, on her desk, were lying a palette and gouache.

This kind of order is natural in Russian because it moves from background/location to new information.

Yes, that is a perfectly natural English translation.

Russian often expresses possession and location differently from English. So:

• у неё на столе лежали... literally = by her on the table lay...
• natural English = she had ... on her desk or there were ... on her desk

So even though the Russian structure is different, that English translation captures the meaning well.

Yes, урок по рисованию is also possible.

Both are understandable, but they differ a little in feel:

• урок рисования = the standard compact way: drawing lesson
• урок по рисованию = more explicit: a lesson in drawing

The version in your sentence, урок рисования, is very common and natural.

Because Russian has no articles.

So:

• палитра can mean a palette or the palette
• гуашь can mean gouache / the gouache

Context tells you which is meant. In this sentence, English would usually say a palette and gouache, but Russian does not need separate words for that.