Learn Hardcore Croatian: Kad sam umorna, učim hrvatski teže nego kad sam odmorena. - When I am tired, I study Croatian harder than when I am rested.

A note on odmorena:
Standard Croatian normally uses the adjective odmoran / odmorna / odmorno.
You will also hear odmoren / odmorena in everyday speech, formed from the verb odmoriti (se). It is common and understandable, though in formal written language odmorna would usually be preferred here:

more standard: kad sam odmorna
very common in speech: kad sam odmorena

What does kad mean here, and is it different from kada?

Kad means “when” (introducing a time clause):

Kad sam umorna = When I am tired
… nego kad sam odmorena = than when I am rested

Kad and kada are basically the same word. The difference is:

kad – shorter, very common in everyday speech and writing
kada – a bit more formal or emphatic; also common

You can say:

Kad sam umorna, učim hrvatski teže…
Kada sam umorna, učim hrvatski teže…

Both are correct; it’s mostly a matter of style and rhythm.

Why is the verb sam (am) repeated: kad sam umorna … kad sam odmorena? Could you leave it out?

Each clause in Croatian needs its own verb, so you normally repeat sam:

Kad sam umorna, učim hrvatski teže nego kad sam odmorena.

You cannot just say:

✗ Kad sam umorna, učim hrvatski teže nego kad odmorena.

That sounds incomplete/wrong.

In very informal spoken language, people sometimes drop repeated verbs if the meaning is crystal clear, but it’s not good to copy this as a learner. In standard Croatian, keep sam in both places.

What exactly is teže? Is it an adjective or an adverb? Why not teži?

Teže here is an adverb in the comparative degree.

Base forms:

teško = hard, with difficulty (adverb)
teže = harder, with more difficulty (comparative adverb)

Adjective forms (used with nouns) are:

težak / teška / teško = difficult, heavy (adjective)
teži / teža / teže = more difficult, heavier (comparative adjective)

In the sentence:

učim hrvatski teže = I study Croatian more difficultly / I find it harder to study Croatian

We are describing how you study (the manner), so we need an adverb, not an adjective.
That’s why it’s teže, not teži.

Why is the word order učim hrvatski teže? Could I say teže učim hrvatski?

Both are possible, and both are correct:

Učim hrvatski teže nego kad sam odmorena.
Teže učim hrvatski nego kad sam odmorena.

They mean the same thing: I study Croatian harder / with more difficulty when I’m tired…

The difference is emphasis:

Učim hrvatski teže… – neutral emphasis, slightly on what you study (Croatian) and then how (more difficultly).
Teže učim hrvatski… – puts a bit more weight on “harder” (The studying is harder).

Croatian word order is fairly flexible, especially with adverbs. As a learner, you can safely use the neutral order:

(Kad sam umorna,) učim hrvatski teže…

Why is there a comma after kad sam umorna?

In Croatian, when a dependent clause (like a kad time clause) comes before the main clause, you normally separate it with a comma:

Kad sam umorna, učim hrvatski teže…
Kad sam odmorena, učim hrvatski lakše.

But if the kad clause comes after the main clause, there is usually no comma:

Učim hrvatski teže kad sam umorna.

So your sentence follows the standard rule:

[Kad sam umorna], [učim hrvatski teže…] → comma in between.

Why is it just hrvatski with no article, and what case is it in?

Croatian has no articles (a, an, the), so you just say:

učim hrvatski = I’m learning Croatian

Here hrvatski is:

the name of the language, and
grammatically, an adjective used as a noun (understood: hrvatski jezik = Croatian language)
in the accusative singular (direct object of učim)

For masculine inanimate nouns/adjectives, the nominative and accusative singular look the same:

Nominative: hrvatski (jezik)
Accusative: učim hrvatski (jezik)

So even though the form doesn’t change, here it functions as the accusative (object) of the verb učiti (to learn).

What does nego do in this sentence, and can it ever be od?

Nego is the regular word for “than” in comparisons with comparatives:

teže nego kad sam odmorena = harder than when I am rested
brže nego prije = faster than before

Use nego after comparative adjectives/adverbs:

veći nego = bigger than
bolji nego = better than
teže nego = harder than

Od can also mean “than” but is used in a different structure:

teže nego prije
vs.
teže od prije

However, in a construction like your sentence, with a full clause after it (…kad sam odmorena), you use nego, not od. So:

✔ teže nego kad sam odmorena
✗ teže od kad sam odmorena (not standard here)

Is there a difference between kad sam odmorena and kad se odmorim?

Yes, they focus on different things:

kad sam odmorena
literally: when I am rested
→ state, condition, how you feel at that time
kad se odmorim
literally: when I rest / when I have rested
→ event, the moment after you finish resting

So:

Učim hrvatski teže kad sam umorna nego kad sam odmorena.
= Comparing two states: tired vs. rested.
Teže učim hrvatski kad sam umorna nego nakon što se odmorim.
= Comparing state of being tired vs. time after the act of resting.

Your original sentence correctly talks about being in a rested state, so kad sam odmorena/odmorna fits well.

Does umorna mean “sleepy,” “tired,” or “exhausted”? Are there other useful words?

Umorna is the general word for tired (physically or mentally).

Related adjectives:

umorna – tired (feminine)
pospana – sleepy (literally: ready to fall asleep)
iscrpljena – exhausted, drained
premorena – overworked, worn out

So:

Kad sam umorna – when I’m tired (in general)
Kad sam pospana – when I’m sleepy
Kad sam iscrpljena – when I’m completely exhausted

In your sentence, umorna is the natural, neutral choice for tired.