Ein Zweig, an dem noch grüne Blätter hängen, liegt jetzt unter dem Ast.

Ein Zweig means a branch / a twig, so it introduces it as something not previously identified.
If the sentence said Der Zweig, it would mean the branch, implying that the branch is already known from context.

In German, this works much like English:

• ein Zweig = a branch
• der Zweig = the branch

Both can be translated as kinds of branch, but they are not exactly the same.

• der Ast usually means a larger branch of a tree
• der Zweig usually means a smaller branch / twig

So in this sentence, the idea is that a smaller branch/twig is lying under the larger branch.

An dem is made of:

• an = a preposition
• dem = the relative pronoun in the dative case

The verb hängen in this sentence is used with an when something is hanging on/from something:

• Blätter hängen an dem Zweig = leaves are hanging on the branch

Because the sentence turns that into a relative clause, dem refers back to Zweig:

• Ein Zweig, an dem noch grüne Blätter hängen, ...
• literally: A branch, on which still green leaves are hanging, ...

So an dem is required by the verb/preposition combination.

The form of the relative pronoun depends not only on the noun’s gender, but also on its case.

Zweig is masculine, so the possible relative pronoun forms are:

• nominative: der
• accusative: den
• dative: dem
• genitive: dessen

Here, the phrase is an dem, and the preposition an in this meaning takes the dative. So the correct form is dem.

A helpful way to think about it:

• gender tells you the set of forms
• the clause tells you which case you need

It gives extra information about Ein Zweig.

Main structure:

• Ein Zweig ... liegt jetzt unter dem Ast.
• A branch ... is now lying under the branch.

Relative clause:

• an dem noch grüne Blätter hängen
• on which green leaves are still hanging

So the full sentence describes which branch we mean: the one that still has green leaves hanging from it.

Because German sends the finite verb to the end in a subordinate clause, and a relative clause is a type of subordinate clause.

Compare:

Main clause:

• Grüne Blätter hängen an dem Zweig.

Relative clause:

• ..., an dem grüne Blätter hängen.

So the verb hängen moves to the end because the clause is introduced by the relative element an dem.

Here noch means still.

So:

• noch grüne Blätter = still green leaves
• or more naturally from the whole clause: leaves that are still green

It suggests that the leaves have not yet dried up, fallen off, or changed color.

In German, noch often means:

• still: Er schläft noch. = He is still sleeping.
• sometimes another / one more, depending on context

Here it clearly means still.

Because grüne Blätter is in the nominative plural and follows no article in that part of the clause.

The noun is Blätter (plural of Blatt), and it is the subject of hängen:

• grüne Blätter hängen = green leaves are hanging

With no article, the adjective takes the strong ending:

• nominative plural: grüne

So:

• grüne Blätter = correct
• grünen Blätter would not fit here

The verb here is simply liegen, which means to lie or to be lying in a horizontal position.

German often distinguishes position verbs more specifically than English:

• liegen = lie, be lying
• stehen = stand, be standing
• sitzen = sit, be sitting
• hängen = hang, be hanging

So:

• Der Zweig liegt unter dem Ast. = The branch is lying under the branch.

You do not need sich here.
And gelegen is a participle used in other constructions, but not the normal choice for this simple sentence.

Because unter is a two-way preposition:

• it takes dative for location
• it takes accusative for movement toward a destination

Here the branch is already located under the larger branch:

• liegt unter dem Ast = location → dative

Compare:

• Der Zweig liegt unter dem Ast. = The branch is lying under the branch.
• Ich lege den Zweig unter den Ast. = I put the branch under the branch.

So:

• unter dem Ast = under the branch, already there
• unter den Ast = to under the branch, movement into that position

Jetzt means now. It tells you that this is the current situation.

• liegt jetzt unter dem Ast = is now lying under the branch

It may suggest a contrast with an earlier state, for example:

• before, the branch was attached to the tree
• now, it is lying underneath

In this sentence, an dem is the normal and natural relative pronoun.

You may see an welchem in more formal or careful language, since welcher / welche / welches can also be used as a relative pronoun. But in everyday German, der/die/das forms are much more common.

So:

• natural: Ein Zweig, an dem noch grüne Blätter hängen, ...
• possible but more formal/literary: Ein Zweig, an welchem noch grüne Blätter hängen, ...

Here it is intransitive.

The subject is:

• grüne Blätter

And the sentence says what they are doing:

• hängen an dem Zweig = are hanging on the branch

There is no direct object.

Compare:

• Die Blätter hängen am Zweig. = intransitive, the leaves are hanging
• Sie hängt das Bild an die Wand. = transitive, she hangs the picture on the wall

German hängen can be used both ways, depending on meaning and structure.

Yes. The core sentence is:

• Ein Zweig liegt jetzt unter dem Ast.

That is a complete sentence.
The relative clause

• an dem noch grüne Blätter hängen

just adds extra descriptive information about the branch.

So the sentence structure is:

• main clause: Ein Zweig liegt jetzt unter dem Ast.
• inserted relative clause describing Zweig

This is very common in German and English alike.