Мне некуда поставить противень, потому что стол уже занят миской с тестом.

Because Russian often uses the dative case to show the person who experiences a situation.

So Мне некуда поставить противень literally means something like:

To me, there is nowhere to put the baking tray.

In more natural English, that becomes:

I have nowhere to put the baking tray.

This is a very common Russian pattern:

• Мне холодно = I am cold
• Мне некогда = I have no time / I’m too busy
• Мне некуда идти = I have nowhere to go

So мне does not mean me in the English sense here; it marks the experiencer.

Некуда means there is nowhere to... or no place to...

In this sentence:

Мне некуда поставить противень
= I have nowhere to put the baking tray

It comes from the -куда idea of direction:

• куда? = where to?
• некуда = nowhere to
• куда поставить? = where to put?
• некуда поставить = nowhere to put

So некуда is about destination/place available for placing or going, not just location in general.

Compare:

• негде = nowhere (to be/do something in a place)
• некуда = nowhere to go/put/move something

This is a very common point of confusion.

• некуда = there is nowhere to...
• никуда = to nowhere / nowhere, usually used with a negative verb

Examples:

• Мне некуда поставить противень.
  = I have nowhere to put the baking tray.

• Я никуда не пойду.
  = I won’t go anywhere.

So in your sentence, некуда is correct because it expresses lack of an available place.

A rough pattern:

• некуда + infinitive
  некуда сесть = nowhere to sit
• никуда + negative finite verb
  никуда не сажусь is not natural here, but structurally this is the idea

Поставить is the perfective verb, and here it fits because we are talking about one completed action: putting the tray down somewhere.

• ставить = imperfective, process/repeated action
• поставить = perfective, single completed placement

So:

Мне некуда поставить противень
means I have nowhere to put/set down the tray as a single act.

If you used ставить, it would sound more like a general or repeated action, which is less natural here.

Compare:

• Куда поставить противень? = Where should I put the tray?
• Я обычно ставлю противень сюда. = I usually put the tray here.

Противень means baking tray, oven tray, or sometimes cookie sheet, depending on context.

Here it is in the accusative case, because it is the direct object of поставить.

For this noun, the accusative singular looks the same as the nominative singular because it is an inanimate masculine noun:

• nominative: противень
• accusative: противень

So:

• противень стоит на столе = the tray is on the table
• поставить противень = to put the tray

Because занят here is the short form of the adjective занятый, and short-form adjectives are commonly used in Russian as part of the predicate, especially for temporary states.

So:

• стол уже занят = the table is already occupied

This is like saying the table is in an occupied state right now.

The full form занятый is more often used as an attribute before a noun:

• занятый стол = an occupied table

But in predicative use, Russian strongly prefers the short form:

• Стол занят. = correct, natural
• Стол занятый. = unnatural in standard usage for this meaning

Because after занят in this meaning, Russian often uses the instrumental case to indicate what something is occupied by.

So:

• стол занят миской
  literally: the table is occupied by a bowl

This is a very common pattern:

• Комната занята гостями. = The room is occupied by guests.
• Стул занят сумкой. = The chair is taken up by a bag.

So миской is instrumental singular of миска.

Because the preposition с here means with, and in this meaning it requires the instrumental case.

So:

• миска = bowl
• тесто = dough
• миска с тестом = a bowl with dough

Case forms:

• nominative: тесто
• instrumental: тестом

This phrase describes the bowl: it is a bowl with dough in it.

Compare:

• чай с молоком = tea with milk
• бутерброд с сыром = a sandwich with cheese
• миска с тестом = a bowl with dough

Russian often does not match English prepositions directly.

In English, we say:

The table is occupied by a bowl of dough.

In Russian, the idea occupied by something is expressed with:

• занят + instrumental

So no separate word for by is needed here. The instrumental case itself carries that function.

That is why Russian says:

стол занят миской с тестом

not something like занят от or занят by-like preposition.

Уже means already.

So:

• стол занят = the table is occupied
• стол уже занят = the table is already occupied

It adds the idea that the table was free before, but now it is not, or that this fact is relevant at the current moment.

In the sentence, it helps explain the problem:

I have nowhere to put the tray, because the table is already occupied by a bowl of dough.

The sentence is natural as written, but Russian word order is more flexible than English word order.

The original:

Мне некуда поставить противень, потому что стол уже занят миской с тестом.

is neutral and natural.

You could move things around for emphasis, for example:

• Стол уже занят миской с тестом, поэтому мне некуда поставить противень.
• Противень мне некуда поставить, потому что стол уже занят миской с тестом.

These versions change the emphasis slightly:

• starting with стол emphasizes the reason first
• starting with противень emphasizes the object that needs placing

But the original is a very normal word order.

Yes. Russian has several natural ways to express the same idea.

For example:

• Мне некуда поставить противень.
• У меня нет места для противня.
• Мне негде поставить противень.
• Стол уже занят миской с тестом, поэтому противень поставить некуда.

These are similar, but not identical:

• некуда поставить focuses on no available destination/place to put it
• негде поставить is also very common in speech and often overlaps in meaning
• нет места focuses more directly on lack of space

Your original sentence is completely natural and idiomatic.

Yes. Потому что is the standard and very common way to say because in everyday Russian.

So:

• ..., потому что стол уже занят...
  = ..., because the table is already occupied...

Other options exist, but they change style or structure:

• так как = more formal, more written
• поэтому = therefore / so, which changes the sentence structure

For example:

• Мне некуда поставить противень, потому что стол уже занят...
• Стол уже занят..., поэтому мне некуда поставить противень.

Both are natural, but the first one directly uses because.