Learn Hardcore German: Ich nehme mein Pausenbrot mit in den Park.

Here mit is not a preposition; it’s the separable prefix of the verb mitnehmen.
As a prefix, mit- adds the meaning “along / with (me/us)” to nehmen.

mitnehmen = take along / take with you
mit (as a preposition) + dative = with (a person/thing), e.g. mit mir, mit dir

In your sentence, you cannot understand mit as a preposition; it doesn’t have its own object and doesn’t trigger a case. It is simply part of the verb mitnehmen.

Why is the word order Ich nehme mein Pausenbrot mit in den Park? Could I also say Ich nehme mein Pausenbrot in den Park mit?

Both sentences are grammatically correct:

Ich nehme mein Pausenbrot mit in den Park.
Ich nehme mein Pausenbrot in den Park mit.

In the first version, mein Pausenbrot mit sticks together more clearly as “I take my snack along”, and in den Park is added as the destination.
In the second version, in den Park stands right next to the verb end mit, which is also very common.

There is no big difference in meaning; both will be understood as “I’m taking my snack with me to the park.” Everyday speech probably favors … in den Park mit a bit.

Why is it in den Park and not im Park or in dem Park?

German uses two different cases with in:

in
- accusative = movement into somewhere (direction)
in
- dative = being in somewhere (location)

Your sentence has movement: you are going into the park with your snack. So you use accusative:

in den Park (accusative) = into the park
im Park = in dem Park (dative, contracted) = in the park (location, no movement)

So Ich nehme mein Pausenbrot in den Park mit focuses on going into the park, not already being in it.

Why is it den Park and not dem Park?

Park is masculine: der Park.
The forms are:

Nominative: der Park
Accusative: den Park
Dative: dem Park

Because we have movement into the park (direction), in takes the accusative, so it must be in den Park, not in dem Park.

What case is mein Pausenbrot, and how do I know?

Mein Pausenbrot is in the accusative case; it’s the direct object of nehme (mitnehmen).

For neuter nouns, nominative and accusative look the same:

Nominative: das Pausenbrot / mein Pausenbrot
Accusative: das Pausenbrot / mein Pausenbrot

You recognize it as accusative here because ich is clearly the subject (nominative), nehme … mit is the verb, and mein Pausenbrot is what is being taken = direct object (accusative).

Why is it mein Pausenbrot and not meinen Pausenbrot?

Pausenbrot is neuter: das Pausenbrot.
For neuter nouns, the possessive mein- does not add an ending in nominative or accusative singular:

Nominative neuter: mein Pausenbrot
Accusative neuter: mein Pausenbrot

You only see -en with masculine accusative singular, e.g. mein Hund → ich sehe meinen Hund.
Since Pausenbrot is neuter, mein Pausenbrot is correct in this sentence.

What exactly does Pausenbrot mean? Is it just “sandwich”?

Literally, Pausenbrot is “break-time bread”:

Pause = break (e.g. school break)
Brot = bread

In practice, Pausenbrot is the snack you take for a break, often a sandwich or slice of bread, sometimes with something on the side (fruit, etc.), especially for school or work.
It’s a typical German school/work-culture word; it doesn’t have a perfect one-word equivalent in English, but “packed lunch / snack / sandwich for the break” are close.

What’s the difference between Ich nehme mein Pausenbrot mit and Ich bringe mein Pausenbrot mit?

Both can translate to “I’m bringing my snack (along)”, but there’s a nuance:

mitnehmen = to take something with you from where you are now to somewhere else (viewed from the starting point).
mitbringen = to bring something along to a place (often viewed from the point of arrival).

In your sentence, if you focus on yourself leaving with the snack, Ich nehme mein Pausenbrot mit in den Park fits well.
If someone is already at the park and asks you to bring your snack with you to them, you’d more naturally say Ich bringe mein Pausenbrot mit in den Park.

How would the position of mit change in a subordinate clause?

In a subordinate clause, the verb goes to the end, and a separable-prefix verb becomes one word again:

Main clause: Ich nehme mein Pausenbrot mit in den Park.
Subordinate clause: …, weil ich mein Pausenbrot mit in den Park nehme.

Here nehme is at the end, and mit stands right before it, still logically attached to the verb.
You do not say weil ich mein Pausenbrot nehme mit.

If I replace mein Pausenbrot with a pronoun (“it”), where does it go?

The pronoun for das Pausenbrot is es. Two common options:

Ich nehme es mit in den Park.
Ich nehme es in den Park mit.

In German, personal pronouns in the middle field usually come early, before longer phrases like in den Park.
So Ich nehme es mit in den Park sounds very natural: subject (ich) – verb (nehme) – pronoun (es) – prefix (mit) – destination (in den Park).

Is Pausenbrot countable? What is the plural, and can I omit the possessive?

Yes, Pausenbrot is countable. The standard plural is:

Singular: das Pausenbrot
Plural: die Pausenbrote

You can omit the possessive if you’re speaking generally:

Ich nehme Pausenbrot mit in den Park. = I take (some) snack/bread for a break to the park.

That sounds a bit generic; in most real contexts you’d specify whose it is:

Ich nehme mein Pausenbrot mit in den Park.
Ich nehme das Pausenbrot mit in den Park. (a specific snack already known from context)