За мостиком начинается узкая тропинка к лесу.

Because the preposition за uses different cases depending on meaning:

• за + instrumental = location: behind / beyond / just past
• за + accusative = motion to a position behind something

So за мостиком means behind / beyond the little bridge as a location.

Compare:

• за мостиком = beyond the bridge
• за мостик = to a position behind the bridge

In this sentence, the path is located there, so Russian uses the instrumental: мостиком.

Yes. Мостик is a diminutive form of мост.

• мост = bridge
• мостик = little bridge, small bridge, footbridge

Diminutives in Russian can show:

• small size
• affection
• a more natural everyday description

Here it most likely just suggests a small bridge or footbridge.

Начинается is the 3rd person singular form of начинаться, which means to begin / to start in an intransitive sense.

In this sentence, the subject is узкая тропинка. So literally:

• Тропинка начинается = The path begins

Russian often uses the reflexive form -ся for things that begin on their own, rather than something starting something else.

Compare:

• Я начинаю урок = I begin the lesson
• Урок начинается = The lesson begins

So начинается is correct because the path itself is what begins.

Because it is the subject of the sentence.

The thing that begins is узкая тропинка:

• тропинка = path, trail
• узкая = narrow

Since тропинка is feminine singular, the adjective must agree with it:

• узкая тропинка

So:

• тропинка = nominative singular feminine
• узкая = nominative singular feminine to match it

Because the preposition к always takes the dative and means to / toward.

• лес = forest
• к лесу = toward the forest / to the forest

Here к лесу describes where the path leads or points.

It does not necessarily mean someone is moving right now. It simply describes the path:

• тропинка к лесу = a path to the forest

Russian word order is flexible, and it often changes the focus rather than the basic meaning.

This sentence starts with За мостиком to set the scene first:

• За мостиком = beyond the little bridge
• then начинается = begins
• then узкая тропинка к лесу = a narrow path to the forest

This order sounds natural when describing a landscape or giving directions.

A different order is possible, for example:

• Узкая тропинка к лесу начинается за мостиком

This is still correct, but it puts more focus on the path itself rather than on the location.

It can mean either in different contexts, but here it most naturally means just past / beyond the little bridge.

In a landscape description, за often means that once you go past something, the next thing is there:

• За домом = behind the house
• За поворотом = around / beyond the turn
• За мостиком = beyond the little bridge

So in this sentence, the best sense is probably just after crossing or passing the little bridge.

Because Russian has no articles.

So узкая тропинка can mean:

• a narrow path
• the narrow path

Which one sounds best depends on context and translation style.

Russian leaves that distinction to context, word order, and the situation.

Yes, начинается is present tense.

In Russian, present tense is often used in descriptions of places and scenes. English may translate it in different natural ways:

• A narrow path to the forest begins beyond the little bridge
• Beyond the little bridge, a narrow path to the forest starts
• There is a narrow path to the forest beginning beyond the little bridge — possible, but less natural

Russian often sounds very straightforward in this kind of descriptive sentence, and English may vary the wording to sound natural.

A fairly literal breakdown is:

• За мостиком = beyond the little bridge
• начинается = begins
• узкая тропинка = a narrow path
• к лесу = to the forest

So the structure is basically:

Beyond the little bridge begins a narrow path to the forest.

That is slightly literary in English, but it matches the Russian structure well.