Через парк проходит новая велодорожка.

Because через means through / across, and it normally takes the accusative case.

• парк = nominative
• через парк = through the park / across the park

This is a very common pattern:

• через город = through the city
• через дорогу = across the road
• через лес = through the forest

So через парк is exactly what you would expect after через.

Because парке would be prepositional case, usually used after prepositions like в or на when talking about location:

• в парке = in the park
• на парке is generally not used here

But this sentence is not using в. It uses через, and через requires the accusative, so you get парк, not парке.

Compare:

• В парке проходит велодорожка. = A bike path runs in the park.
• Через парк проходит велодорожка. = A bike path runs through the park.

The meaning is slightly different.

Here проходит means something like:

• runs through
• goes through
• passes through

In Russian, проходить can be used not only for people physically walking, but also for things like:

• roads
• paths
• borders
• lines
• routes

So in this sentence, новая велодорожка проходит через парк means the bike path’s route goes through the park.

Similar examples:

• Через деревню проходит дорога. = A road runs through the village.
• Граница проходит по реке. = The border runs along the river.

Because велодорожка is the subject of the sentence.

The sentence structure is:

• Через парк = through the park
• проходит = runs / goes
• новая велодорожка = a new bike path

If you rearrange it into a more English-like order, it becomes:

• Новая велодорожка проходит через парк.

Now it is easier to see that велодорожка is the thing doing the action, so it stays in the nominative.

Because новая agrees with велодорожка.

Велодорожка is:

• feminine
• singular
• nominative

So the adjective must also be:

• feminine
• singular
• nominative

That gives новая.

Compare:

• новый парк = a new park
• новая велодорожка = a new bike path
• новое здание = a new building
• новые дома = new houses

This is standard adjective-noun agreement in Russian.

Yes, it is a normal modern word meaning bike path or cycle lane/path.

It is made from:

• вело- = bicycle-related
• дорожка = little road / path / track

So literally it is something like bicycle path.

Related words:

• велосипед = bicycle
• дорожка = path, track
• беговая дорожка = running track / treadmill
• веломаршрут = bike route

A learner should just treat велодорожка as a standard vocabulary word.

Russian word order is flexible, and the beginning of the sentence often highlights what the speaker wants to focus on.

These are both natural:

• Через парк проходит новая велодорожка.
• Новая велодорожка проходит через парк.

The difference is mostly one of emphasis:

• Через парк проходит новая велодорожка puts early attention on through the park
• Новая велодорожка проходит через парк starts with the subject, which may feel more neutral to English speakers

So the original sentence is not strange; it is just using Russian word order to foreground the location/path.

Not by itself.

Проходит here describes the route or location of the bike path: it runs through the park.

It does not directly mean is being built.

If you wanted to say is being built, Russian would use something like:

• Через парк строят новую велодорожку. = They are building a new bike path through the park.
• В парке строят новую велодорожку.
• Новая велодорожка строится через парк. = A new bike path is being built through the park.

So проходит is about where it goes, not about construction.

Проходит is:

• 3rd person singular
• present tense
• from the verb проходить

That matches the singular subject велодорожка.

Basic forms:

• я прохожу = I pass / go through
• ты проходишь = you pass / go through
• он/она проходит = he/she/it passes / goes through
• они проходят = they pass / go through

Here велодорожка is singular, so the verb is singular: проходит.

Russian has no articles like English a and the.

So новая велодорожка can mean:

• a new bike path
• the new bike path

Which one is meant depends on context.

In many isolated example sentences, English translations use a because the noun is being introduced for the first time. But Russian itself does not mark that distinction with an article.

Yes, definitely.

проходить через + accusative is a very useful pattern meaning:

• to go through
• to pass through
• to run through

Examples:

• Поезд проходит через тоннель. = The train goes through the tunnel.
• Тропа проходит через лес. = The trail runs through the forest.
• Через город проходит река. = A river runs through the city.

So this sentence is a good model for a common and natural Russian structure.