Learn Hardcore Russian: У нас есть время, чтобы немного погулять в парке. - We have time to take a short walk in the park.

у нас literally means at us / by us (the possessor is expressed as a location).
есть время means there is time (available).
So У нас есть время = We have time (i.e., time is available to us).

What case is нас in, and why?

нас is genitive plural of мы (we). After у (meaning at / by / in someone’s possession), Russian uses the genitive case:

у меня, у тебя, у него, у нас, etc.

Is есть always required? Can it be omitted?

Often it can be omitted in the present tense:

У нас (есть) время.
Both are possible, but the nuance changes slightly:
With есть, it sounds a bit more explicit: we do have / there is indeed time (availability is emphasized).
Without есть, it can sound more neutral or conversational.

Why is there a comma before чтобы?

Because чтобы introduces a subordinate clause (a purpose/result clause). In Russian, subordinate clauses are normally separated by a comma:

У нас есть время, чтобы…
The comma marks the boundary between the main clause and the чтобы clause.

What does чтобы mean here, and how is it different from что?

Here чтобы means in order to / so that and is used to introduce a purpose:

…время, чтобы погулять… = time to (in order to) take a walk…

что is usually that/which and introduces content/relative clauses, not purpose:

Я знаю, что… = I know that…

Why do we use an infinitive after чтобы here?

When the subject is the same in both clauses, Russian commonly uses чтобы + infinitive:

У нас есть время, чтобы погулять… (We have time to take a walk.)

If you explicitly state a different subject, you usually use a finite verb:

У нас есть время, чтобы он погулял. = We have time for him to take a walk.

What does немного modify, and can it move in the sentence?

немного means a little / for a bit and here it modifies the action погулять (how long / to what extent you walk).
It’s fairly mobile:

…чтобы немного погулять в парке. (most common)
…чтобы погулять немного в парке. (also possible; slightly more “afterthought” feel)

Why is it погулять and not гулять? (Aspect question)

погулять is typically perfective with the meaning to take a walk for a while / have a walk (as a single event). It fits well with немного and the idea of “we have time to do this activity.”
гулять is imperfective and is more about the process/habit:

Мы любим гулять в парке. = We like walking in the park. In this sentence, the goal is one walk (for a bit), so погулять is natural.

Why is it в парке and not в парк?

Because в + prepositional is used for location (in/at the park):

в парке = in the park (prepositional case)

в + accusative is used for motion toward a place (into/to the park):

в парк = to the park (accusative case)

Here the walking happens in the park, not just going to it.

What case is время, and why doesn’t it change after есть?

время is nominative (or sometimes analyzed as the subject-like noun) in the existential/possessive pattern у X есть Y.
Even though English treats time as an object in we have time, Russian treats время as the thing that “exists/ is available,” so it stays in its basic form:

есть что? время (nominative)

In negative sentences, you’ll often see genitive:

У нас нет времени. = We don’t have time.