Sie stützt sich auf den Tisch und trifft dabei mit dem Ellbogen die Tasse.

Here it can only mean she.

Why:

• sie = she or they
• formal Sie = you
• But the verb is stützt, which is 3rd person singular

So:

• she supports herself / leans → sie stützt
• they support themselves / lean → sie stützen
• you support yourself / lean (formal) → Sie stützen

Because the verb is singular, Sie in this sentence must mean she.

Because sich auf etwas stützen is a reflexive expression.

It means:

• to support oneself on something
• to lean on something
• to brace oneself against/on something

Without sich, stützen usually means to support something:

• Er stützt die Leiter. = He supports the ladder.
• Sie stützt sich auf den Tisch. = She supports herself on the table.

So sich shows that the subject is doing the action to herself.

Because auf is a two-way preposition, and in sich auf etwas stützen, German normally uses the accusative.

So:

• auf den Tisch = onto / against / on the table as the target of the support
• auf dem Tisch = on the table in a location sense

A helpful contrast:

• Das Buch liegt auf dem Tisch. = The book is on the table.
• Sie stützt sich auf den Tisch. = She leans/supports herself on the table.

Even though English does not show this difference clearly, German treats the table here as the thing she is placing her weight onto, so accusative is standard.

You could, but it would not mean exactly the same thing.

• sich auf den Tisch stützen = to brace/support oneself on the table, often with the hands or arms
• sich an den Tisch lehnen = to lean against the table
• sich auf den Tisch lehnen = to lean onto the table, often with the upper body

So stützen focuses more on using the table as support, while lehnen focuses more on leaning.

In your sentence, stützt sich auf den Tisch is a very natural choice if she is putting weight on the table and, while doing that, hits the cup.

Because German often omits the repeated subject in coordinated main clauses when it is the same as in the first clause.

So these are both possible:

• Sie stützt sich auf den Tisch und trifft dabei mit dem Ellbogen die Tasse.
• Sie stützt sich auf den Tisch und sie trifft dabei mit dem Ellbogen die Tasse.

The second version is grammatical, but the first is more natural and less heavy because the subject is already clear.

dabei means something like:

• while doing so
• in the process
• at the same time

It connects the second action to the first one.

So the sentence is not just saying:

1. she leans on the table
2. she hits the cup

It is saying that the second action happens as part of the first action.

Without dabei, the sentence would still be grammatical, but dabei makes the link between the two actions clearer.

No. Here trifft means hits, strikes, or bumps into.

The verb treffen has different meanings depending on context:

• jemanden treffen = to meet someone
• ein Ziel treffen = to hit a target
• die Tasse treffen = to hit the cup

In this sentence, it is clearly the physical meaning: her elbow makes contact with the cup.

Because German often uses the definite article with body parts when it is already obvious whose body part is meant.

So German naturally says:

• mit dem Ellbogen
• an der Hand
• ins Gesicht

Where English often prefers:

• with her elbow
• on his hand
• in the face

You can say mit ihrem Ellbogen, but that usually adds emphasis, contrast, or clarity.

For example:

• Sie traf die Tasse mit ihrem Ellbogen, nicht mit ihrer Hand.

In your sentence, mit dem Ellbogen is the normal idiomatic choice.

They have different jobs in the sentence.

mit dem Ellbogen

The preposition mit always takes the dative.

So:

• mit dem Ellbogen
• mit der Hand
• mit einem Messer

die Tasse

This is the direct object of trifft, so it is in the accusative.

Because Tasse is feminine, the article is:

• nominative singular: die
• accusative singular: die

So it looks the same in both cases, but here it is functioning as the accusative object.

Because German word order is flexible, and this order is very natural.

In the second clause:

• trifft = verb
• dabei = adverb
• mit dem Ellbogen = prepositional phrase
• die Tasse = direct object

Putting die Tasse at the end makes it sound like the final piece of information, which works well here.

German could also allow other orders, for example:

• ... und trifft dabei die Tasse mit dem Ellbogen.

But ... trifft dabei mit dem Ellbogen die Tasse sounds very natural and smooth.

Yes. Both mean elbow.

• Ellenbogen is the fuller and more traditional form
• Ellbogen is a shorter variant

Both are understood and accepted. In many contexts, Ellenbogen is a bit more common in standard German, but Ellbogen is also normal.

So in your sentence, mit dem Ellbogen is perfectly fine.

No. treffen does not have to mean an intentional hit.

In this sentence, it very likely means she accidentally knocks into the cup with her elbow while leaning on the table.

German uses treffen for both intentional and unintentional contact; the context tells you which is meant. Here, the presence of dabei strongly suggests an accidental action that happens in the process.