Learn Hardcore Croatian: Kad osjetim sumnju, sjedim u parku i mirno učim hrvatski. - When I feel doubt, I sit in the park and calmly study Croatian.

osjetiti → osjetim (perfective): to feel / sense at a specific moment, to notice a feeling.
osjećati → osjećam (imperfective): to feel something as an ongoing state.

So:

Kad osjetim sumnju = When I (suddenly) feel / detect doubt (the moment I notice it).
Kad osjećam sumnju = When I (generally) feel doubt (more like a lasting state).

Both are grammatically fine, but osjetim focuses on the moment you become aware of the doubt.

What case is sumnju and why is it used?

Sumnju is in the accusative singular.

The verb osjetiti / osjećati takes a direct object:

nominative: sumnja (doubt – as a subject)
accusative: sumnju (doubt – as an object)

So:

Sumnja je jaka. – The doubt is strong. (subject → nominative)
Osjetim sumnju. – I feel doubt. (object → accusative)

In the sentence Kad osjetim sumnju, sumnju is what you feel, so it must be accusative.

What case is u parku, and why not u park?

U parku uses the locative case (singular) with u to express location:

u parku = in the park (where you are – static location)

Croatian uses u + locative for where? and u + accusative for where to? / into where?

Compare:

Sjedim u parku. – I am sitting in the park. (location → locative)
Idem u park. – I am going to the park. (direction → accusative)

So sjedim u parku is correct because you are already there, just sitting.

Why is there a comma after Kad osjetim sumnju?

Kad osjetim sumnju is a dependent clause (a time clause). Croatian normally separates such a clause from the main clause with a comma when it comes first:

Kad osjetim sumnju, sjedim u parku i mirno učim hrvatski.

If you reverse the order, you usually do not use a comma:

Sjedim u parku i mirno učim hrvatski kad osjetim sumnju.

So the comma marks the boundary between the when-clause and the main action.

Why is ja (I) not written? Is it optional?

Yes, ja is optional here.

Croatian is a pro‑drop language: the verb ending already shows the person:

osjetim = I feel
sjedim = I sit
učim = I study

So you normally say:

Kad osjetim sumnju… (without ja)

You can add ja for emphasis or contrast:

Kad ja osjetim sumnju, sjedim u parku. – When I (as opposed to others) feel doubt…

Can the word order at the end change? For example, can I say mirno hrvatski učim?

The neutral, natural order here is:

mirno učim hrvatski

Other orders are possible, but they change emphasis or sound odd:

Učim mirno hrvatski. – possible, but puts a bit more weight on učim and can sound slightly marked.
Mirno hrvatski učim. – grammatically possible, but sounds poetic / unusual in everyday speech.

The usual pattern is:

Verb: učim
Adverb: mirno (how?)
Object: hrvatski (what?)

or adverb first for slight emphasis:

Mirno učim hrvatski.

The original sjedim u parku i mirno učim hrvatski sounds the most natural.

What exactly does mirno mean here? Is it “quietly” or “peacefully”? How is it different from tiho?

Mirno is the adverb from miran and usually means:

calmly, peacefully, without stress, without disturbance.

In this sentence it suggests you study in a calm, relaxed way, maybe also in a quiet environment.

Tiho specifically means:

quietly, in a low volume (not loud).

So:

mirno učim hrvatski – I study in a calm, unhurried, peaceful way.
tiho učim hrvatski – I study quietly (not making noise), but it doesn’t necessarily say anything about your inner state.

Why is hrvatski not capitalized? In English “Croatian” is.

In Croatian, names of languages are written with a lowercase initial letter:

hrvatski – Croatian
engleski – English
njemački – German

You only capitalize Hrvatski when it is part of a proper name (e.g. Hrvatski sabor – Croatian Parliament) or at the beginning of a sentence.

So učim hrvatski is correctly written with a lowercase h.

Is hrvatski here an adjective or a noun? Why can you just say učim hrvatski without jezik?

Grammatically, hrvatski is an adjective, but in this context it functions like a short form for “Croatian language”.

You can say:

učim hrvatski – I am learning Croatian.
učim hrvatski jezik – I am learning the Croatian language.

Both are correct and mean the same thing. Adding jezik (language) is slightly more explicit or formal, but in everyday speech people usually just say učim hrvatski.

Why present tense everywhere? Could you use future tense instead?

The present tense here expresses a general, repeated situation (habit):

Kad osjetim sumnju, sjedim u parku i mirno učim hrvatski.
= Whenever I feel doubt, I (usually) sit in the park and study Croatian.

This is similar to English present simple in “When I feel doubt, I sit in the park and study Croatian.”

You could use future if you are talking about a specific future pattern, e.g.:

Kad osjetim sumnju, sjedit ću u parku i mirno učiti hrvatski.
= When I (will) feel doubt, I will sit in the park and calmly study Croatian.

But for general habits and routines, the present is the normal choice.

Is učim se hrvatski correct, like “I’m teaching myself Croatian”?

In standard Croatian, učiti se does not work the same way as English “learn/teach oneself” here.

Correct ways to say it:

Učim hrvatski. – I am learning Croatian.
Sam učim hrvatski. – I am learning Croatian by myself.
Samostalno učim hrvatski. – I am learning Croatian independently.

Učiti se is used more in senses like:

Djeca se u školi uče matematici. – Children are taught math at school.

So you should say učim hrvatski, not učim se hrvatski, when you mean I am learning Croatian.