Die Wurzel, über die ich fast gestolpert wäre, kommt direkt vor der Hecke aus dem Boden.

Because that part is a relative clause. It gives extra information about die Wurzel.

• Die Wurzel = the noun being described
• über die ich fast gestolpert wäre = the clause describing it

In German, relative clauses are normally set off with commas, so you get:

Die Wurzel, über die ich fast gestolpert wäre, ...

English often does something similar with: The root, which I almost tripped over, ...

Here, die is a relative pronoun, not just the article the.

It refers back to die Wurzel.

So:

• die Wurzel = feminine singular
• the relative pronoun must match that noun in gender and number
• but its case depends on its role inside the relative clause

That is why you get die here.

Because stolpern über takes the accusative.

So the logic is:

• über can take different cases in different situations
• but with the verb stolpern, German normally uses the fixed pattern über + accusative
• therefore the relative pronoun is accusative: die

Compare:

• Ich stolpere über die Wurzel.
• Die Wurzel, über die ich stolpere, ...

So die here is accusative feminine singular.

Because this is a subordinate clause: a relative clause. In German subordinate clauses, the finite verb goes to the end.

The basic main-clause version would be:

• Ich wäre fast über die Wurzel gestolpert.

But inside the relative clause, the verb moves to the end:

• ..., über die ich fast gestolpert wäre, ...

Also notice that with a two-part verb form, the past participle comes before the auxiliary at the end:

• gestolpert wäre

Because ich fast gestolpert wäre expresses something that almost happened but did not actually happen.

German often uses Konjunktiv II in the past for this kind of unreal past idea:

• Ich wäre fast gestolpert. = I almost would have tripped / I almost tripped.

So wäre here signals an unrealized past event.

If you said ich bin fast gestolpert, that can also exist in everyday German, but ich wäre fast gestolpert is a very standard way to express I almost tripped.

Because the perfect forms of stolpern are usually built with sein, not haben.

So:

• ich bin gestolpert
• ich wäre gestolpert

This is common with many intransitive verbs connected with movement or a change of state. In modern standard German, stolpern normally goes with sein.

In the equivalent main clause, you would say:

• Ich wäre fast gestolpert.

When that becomes a subordinate clause, the verb material moves to the end, but fast stays in front of the participle:

• ..., über die ich fast gestolpert wäre, ...

So the order reflects normal German subordinate-clause word order. Fast modifies the whole event: the tripping almost happened.

Because vor is a two-way preposition:

• dative for location: where?
• accusative for direction/movement toward a destination: where to?

Here the sentence describes a location:

• the root comes out of the ground in front of the hedge

So German uses the dative:

• vor der Hecke

Since die Hecke is feminine, its dative form is der Hecke.

Because aus always takes the dative.

• der Boden = the ground/soil
• dative singular = dem Boden

So:

• aus dem Boden = out of the ground

This is a very common expression for something emerging from the earth.

The main clause is still:

• Die Wurzel kommt direkt vor der Hecke aus dem Boden.

The relative clause is just inserted after Die Wurzel:

• Die Wurzel, über die ich fast gestolpert wäre, kommt direkt vor der Hecke aus dem Boden.

German still follows the verb-second rule in the main clause. The finite verb of the main clause is kommt, and it remains in the second main-clause position. The relative clause does not change that; it is just additional information inserted between the subject and the verb.

Yes, but in this kind of sentence direkt often means something like:

• right in front of
• immediately in front of

So direkt vor der Hecke means the root is very close to the hedge, not merely somewhere in front of it.

It adds emphasis to the exact location.

Yes. German allows several natural alternatives, for example:

• Die Wurzel, über die ich beinahe gestolpert wäre, kommt direkt vor der Hecke aus dem Boden.
• Die Wurzel kommt direkt vor der Hecke aus dem Boden, und ich wäre fast über sie gestolpert.

A couple of notes:

• beinahe and fast are very similar here.
• In the original sentence, the relative clause makes the root sound very specifically identified: the root that I almost tripped over.

So the original wording is perfectly natural, especially if the speaker is pointing out a particular root.