Der Arzt schaut kurz auf mein Kinn und sagt, dass alles in Ordnung ist.

It has two main clauses joined by und, plus one subordinate clause:

• Der Arzt schaut kurz auf mein Kinn
• (der Arzt) sagt
• dass alles in Ordnung ist

So the pattern is:

main clause + und + main clause + subordinate clause

The dass clause depends on sagt and tells you what the doctor says.

Because der Arzt is the subject of the sentence — the person doing the actions schaut and sagt.

• der Arzt = nominative masculine singular
• den Arzt = accusative masculine singular

Here, the doctor is doing the looking and speaking, so German uses the nominative case:

• Der Arzt schaut ...
• Der Arzt sagt ...

If the doctor were the object, you would get den Arzt instead.

Because the subject stays the same, German can leave it out in the second clause:

• Der Arzt schaut kurz auf mein Kinn und sagt ...

This means:

• The doctor looks ... and says ...

You could repeat der Arzt, but it would sound heavier and less natural in a simple sentence like this.

So this is a normal case of subject ellipsis in coordinated clauses.

kurz is an adverb. Here it means something like briefly or for a moment.

It modifies schaut:

• schaut kurz = looks briefly

So the sentence is not just saying that the doctor looks at the chin, but that he does so only for a short time.

Because the verb phrase here is auf etwas schauen, meaning to look at something.

So:

• auf = the preposition used with this expression
• mein Kinn = the thing being looked at

Also, Kinn is a neuter noun:

• nominative: das Kinn
• accusative: das Kinn

With mein, the accusative neuter form is still mein:

• auf mein Kinn

So even though English just says at my chin, German uses auf + accusative here.

Because auf here takes the accusative, not the dative.

German prepositions like auf can sometimes take different cases, but with schauen auf the thing you are looking at is normally in the accusative:

• auf mein Kinn

Since Kinn is neuter, the accusative form with mein is:

• mein Kinn

Not:

• meinem Kinn — that would be dative

So this is a case question, not just a vocabulary question.

Yes. German has several ways to express look at.

Possible alternatives include:

• Der Arzt schaut mein Kinn kurz an.
• Der Arzt sieht sich mein Kinn kurz an.
• Der Arzt sieht kurz auf mein Kinn. (less common than schaut in this use)

A useful difference:

• auf etwas schauen = direct your gaze toward something
• etwas anschauen / ansehen = look at something

For learners, anschauen and ansehen often feel closer to the English pattern look at.

Because dass introduces a subordinate clause, and in German subordinate clauses are normally separated by a comma.

So:

• ..., sagt, dass alles in Ordnung ist.

That comma marks the beginning of the clause that depends on sagt.

This is a very common pattern in German:

• Er sagt, dass ...
• Ich glaube, dass ...
• Sie weiß, dass ...

Because dass is a conjunction meaning that.

Compare:

• dass = conjunction
• das = article or pronoun

Examples:

• Ich weiß, dass er kommt.
• Das Buch ist neu.

In your sentence, dass introduces what the doctor says, so the conjunction form is needed:

• ..., dass alles in Ordnung ist.

Because dass sends the conjugated verb to the end of the subordinate clause.

Main clause word order:

• Alles ist in Ordnung.

Subordinate clause with dass:

• dass alles in Ordnung ist

This is one of the most important German word-order rules: in many subordinate clauses, the finite verb moves to the end.

Because und is simply joining two main clauses, and German usually does not put a comma before und in that situation.

So this is normal:

• Der Arzt schaut kurz auf mein Kinn und sagt ...

The comma in the sentence is not connected to und. It is there because of dass:

• ..., sagt, dass ...

So the rule here is:

• no comma before und
• comma before the dass-clause

It is a very common expression in German.

• alles = everything
• in Ordnung = in order / okay / fine

So alles in Ordnung ist means something like:

• everything is fine
• everything is okay

Also, Ordnung is capitalized because it is a noun. German capitalizes nouns even inside fixed expressions.

Because they are all nouns, and German capitalizes all nouns.

So:

• der Arzt
• mein Kinn
• in Ordnung

This is one of the most visible spelling differences between German and English. Even when a noun appears inside a phrase like in Ordnung, it still keeps its capital letter.