Der Ast, auf dem der Vogel sitzt, bewegt sich im Wind, und ein Zweig fällt fast auf den Weg.

This part is a relative clause describing der Ast.

• dem refers back to der Ast
• it matches Ast in gender and number: masculine singular
• its case comes from its role inside the relative clause

Because the bird is sitting on the branch, auf expresses a location, so German uses the dative: auf dem. If there were movement onto the branch, you would expect auf den instead.

Because this is a relative clause, and in German subordinate clauses normally send the conjugated verb to the end.

So:

• main clause: Der Ast bewegt sich im Wind.
• relative clause: auf dem der Vogel sitzt

That is why sitzt comes last in that clause.

Because German sets off relative clauses with commas. The clause auf dem der Vogel sitzt gives extra information about der Ast, so it must be enclosed by commas.

So the structure is:

Der Ast, auf dem der Vogel sitzt, bewegt sich im Wind.

Those two commas are required.

That comma is different from the commas around the relative clause.

In German, a comma before und joining two main clauses is often optional. Here it helps separate the long first clause from the second one:

Der Ast ... bewegt sich im Wind, und ein Zweig fällt ...

So:

• commas around the relative clause: required
• comma before und: often stylistic/optional, used for clarity

Because sich bewegen is the normal intransitive German expression for to move.

• bewegen = to move something
• sich bewegen = to move, to be moving

So:

• Der Wind bewegt den Ast. = The wind moves the branch.
• Der Ast bewegt sich. = The branch moves.

The sich is an important part of the verb here.

im is just the common contraction of in dem.

So:

• im = in dem

German uses these contractions very often, especially with common prepositions and articles. im Wind sounds more natural than the full in dem Wind here.

Because auf is a two-way preposition in German:

• dative for location: auf dem Ast = on the branch
• accusative for direction/movement toward something: auf den Weg = onto the path / toward the path

In this sentence, the bird is already on the branch, so German uses auf dem. The twig is falling toward the path, so German uses auf den Weg.

They are similar, but not identical.

• Ast usually means a larger branch/bough
• Zweig usually means a smaller branch/twig

So the sentence paints a natural picture: the bird is on a bigger branch, while a smaller branch or twig is the one that falls.

fast means almost.

In ein Zweig fällt fast auf den Weg, it means the twig almost falls onto the path or comes very close to landing on the path.

German adverbs like fast often appear before the part they are most closely connected to. Here it stands before auf den Weg, which makes the sentence sound natural and clearly shows what is being limited: the fall in relation to the path.

Because after und, you normally get another main clause, and main clauses in German follow the verb-second rule.

In the second clause:

• first position: ein Zweig
• second position: fällt

So und ein Zweig fällt ... is the normal word order.

You can put something else first for emphasis, but then the verb still has to stay second, for example: ... und fast fällt ein Zweig auf den Weg. That would sound more marked and emphatic.

Yes. auf welchem der Vogel sitzt is grammatically correct.

But there is a style difference:

• auf dem der Vogel sitzt = normal and common
• auf welchem der Vogel sitzt = more formal or literary

Most everyday German would prefer auf dem.

You may hear that in some regional or colloquial German, but it is not the standard choice here.

Standard German prefers the preposition + relative pronoun:

Der Ast, auf dem der Vogel sitzt

That is because the verb idea requires auf, and standard German keeps that preposition in the relative clause. So for learners, auf dem is the form to use.