Heute habe ich keinen Appetit auf Kuchen.

German main clauses usually follow the verb-second rule: the finite verb must be in the second position.

So in:

Heute habe ich keinen Appetit auf Kuchen.

• Heute takes the first position
• habe must come second
• ich comes after the verb

Compare:

• Ich habe heute keinen Appetit auf Kuchen.
• Heute habe ich keinen Appetit auf Kuchen.

Both are correct. Starting with Heute gives today a bit more emphasis.

Because once something other than the subject is placed first, German usually puts the subject after the finite verb.

So:

• Ich habe ... = normal subject-first order
• Heute habe ich ... = time expression first, so the verb stays second and the subject moves after it

This is often called inversion in beginner explanations, though it is really just the normal German word-order rule.

Because Appetit is masculine and it is the direct object of haben, so it is in the accusative case.

The negative article kein changes like ein:

• nominative masculine: kein Appetit
• accusative masculine: keinen Appetit

Since haben takes an accusative object, you get:

• Ich habe keinen Appetit.

Use kein to negate a noun that would otherwise often have ein or no article.

Here, what is being negated is Appetit:

• Ich habe Appetit. = I have an appetite.
• Ich habe keinen Appetit. = I have no appetite.

By contrast, nicht usually negates:

• a verb
• an adjective
• an adverb
• a whole clause
• or something with a definite article

So Ich habe nicht Appetit is not natural German.

Appetit auf + accusative is a fixed expression in German. It means an appetite for or a desire for something.

Examples:

• Appetit auf Pizza
• Appetit auf Eis
• Appetit auf Kuchen

You should learn it as a chunk:

Appetit auf etwas haben

Similarly, German also often uses:

• Lust auf etwas haben = to feel like something

Here it is accusative, because the expression is Appetit auf + accusative.

So in a phrase with an article, you can see the case more clearly:

• auf einen Kuchen (accusative masculine)
• auf den Kuchen (also accusative masculine, but same form as nominative)

In your sentence, Kuchen has no article, so its form does not visibly change.

Because Kuchen is being used in a general sense here: cake as a type of food, not a specific cake.

So:

• Appetit auf Kuchen = appetite for cake, generally
• Appetit auf den Kuchen = appetite for the cake, a specific cake

This works a lot like English:

• I’m in the mood for cake
• I’m in the mood for the cake

Yes. That is also completely correct.

Compare:

• Heute habe ich keinen Appetit auf Kuchen.
• Ich habe heute keinen Appetit auf Kuchen.

The meaning is basically the same, but the focus changes slightly:

• Heute ... puts more emphasis on today
• Ich ... is a more neutral starting point

German often changes word order to shift emphasis without changing the basic meaning much.

Not necessarily. It specifically says that you do not have an appetite for cake.

It can imply:

• you are not interested in cake today
• maybe you are not hungry in general
• or maybe you would eat something else, just not cake

If you want to say you have no appetite at all, you could simply say:

• Ich habe keinen Appetit.

They overlap, but they are not exactly the same.

• Appetit = appetite, often for a particular food
• Hunger = hunger in the physical sense
• Lust auf etwas = feeling like something, desire for something

Examples:

• Ich habe Hunger. = I am hungry.
• Ich habe Appetit auf Kuchen. = I feel like eating cake / I have an appetite for cake.
• Ich habe Lust auf Kuchen. = I feel like having cake.

In your sentence, Appetit auf Kuchen sounds very natural because it is about a specific food craving or appetite.

Yes. German time expressions are flexible.

For example:

• Heute habe ich keinen Appetit auf Kuchen.
• Ich habe heute keinen Appetit auf Kuchen.

Both are natural.

Putting heute first makes it more prominent. Putting it later sounds a bit more neutral. In both cases, the sentence is correct.

Yes. The pattern is very useful:

Heute habe ich keinen Appetit auf + food

You can swap in other foods:

• Heute habe ich keinen Appetit auf Pizza.
• Heute habe ich keinen Appetit auf Schokolade.
• Heute habe ich keinen Appetit auf Fleisch.

You can also make it positive:

• Heute habe ich Appetit auf Kuchen.

So this is a good sentence to memorize as a model.