Wij lopen langs de bibliotheek naar het park.

Both are possible.

• wij is the full/stressed form of we
• we is the unstressed form and is very common in everyday Dutch

So this sentence could also be:

• We lopen langs de bibliotheek naar het park.

In many situations, Dutch speakers would naturally say we.
Wij can sound a bit more emphatic, contrastive, or simply slightly more formal/careful.

Because the subject is wij (we), and in the present tense Dutch uses lopen with wij / we.

Present tense of lopen:

• ik loop — I walk
• jij loopt — you walk
• hij/zij loopt — he/she walks
• wij lopen — we walk
• jullie lopen — you all walk
• zij lopen — they walk

So Wij lopen is the normal present-tense form.

Its basic meaning is to walk or to go on foot.

In this sentence, lopen clearly means walking:

• Wij lopen langs de bibliotheek naar het park.
  = We are walking / We walk past the library to the park.

Dutch also has gaan, which means to go in a more general sense.
So:

• Wij gaan naar het park. = We are going to the park.
• Wij lopen naar het park. = We are walking to the park.

The second one specifically tells you how you are going.

Here langs means something like:

• past
• by
• along

So langs de bibliotheek means that the route goes past the library or by the library.

It does not mean that the library is the destination. The library is just something you pass on the way.

Because the two prepositions express different ideas:

• naar = movement toward a destination
• langs = movement past / alongside something

So in this sentence:

• de bibliotheek is something you pass
• het park is where you are going

That is why Dutch says:

• langs de bibliotheek — past the library
• naar het park — to the park

Because Dutch nouns take one of two main definite articles:

• de
• het

You usually have to learn which noun goes with which article.

In this sentence:

• de bibliotheek — the library
• het park — the park

A useful general rule:

• most nouns are de words
• many neuter nouns are het words

But for individual nouns, it is often best just to memorize the article together with the noun:

• de bibliotheek
• het park

Dutch normally keeps the article here, just like English does in to the park.

So:

• naar het park = to the park

Leaving out the article would usually sound ungrammatical in standard Dutch.

This is very common with places:

• naar de winkel — to the shop
• naar de school — to the school
• naar het station — to the station

Yes, that is possible, but it shifts the focus a little.

• Wij lopen langs de bibliotheek naar het park.
  This presents the route first and then the destination.

• Wij lopen naar het park langs de bibliotheek.
  This gives the destination first, and then adds extra route information.

Both can be understandable, but the original sentence is very natural if you want to describe the path in order: first you pass the library, then you reach the park.

Yes. It is a prepositional phrase:

• langs = preposition
• de bibliotheek = noun phrase

Together they function as one unit: langs de bibliotheek.

The same is true for naar het park:

• naar = preposition
• het park = noun phrase

So the sentence structure is roughly:

• Wij — subject
• lopen — verb
• langs de bibliotheek — route phrase
• naar het park — destination phrase

It can mean different things depending on context.

The Dutch present tense often covers:

• we walk
• we are walking
• sometimes even we are going to walk / we will walk

So Wij lopen langs de bibliotheek naar het park could mean:

• a present action happening now
• a habitual action
• a near-future plan, if the context makes that clear

Dutch often relies on context more than English does here.

Yes, but very often they would say We lopen... instead.

So in everyday spoken Dutch, this is probably more common:

• We lopen langs de bibliotheek naar het park.

The version with wij is still correct; it just sounds a bit fuller or more emphatic.