Learn Hardcore Finnish: Kun luen uutiskirjettä, merkitsen yhden vaikean sanan muistikirjaan. - When I read the newsletter, I mark one difficult word in my notebook.

Kun is a subordinating conjunction meaning when (also sometimes as/while depending on context). It introduces a time clause: Kun luen uutiskirjettä = When I read the newsletter.
The main clause is merkitsen yhden vaikean sanan muistikirjaan. Finnish often puts the time clause first, and then separates it with a comma.

Why is there a comma after uutiskirjettä?

In Finnish, a subordinate clause introduced by words like kun, että, koska, jos is typically separated from the main clause with a comma.
So: Kun luen uutiskirjettä, merkitsen... is standard punctuation.

Why is luen in the present tense if this could refer to the future or repeated action?

Finnish present tense covers:

what you do now
what you do habitually
sometimes what you will do in a scheduled/typical situation

Here it’s most naturally a habitual meaning: Whenever/When I read a newsletter, I note down... Finnish doesn’t need a special tense for this.

Why isn’t the subject minä written? How do you know who is doing it?

Finnish usually drops subject pronouns because the verb ending shows the person.

luen = I read
merkitsen = I mark/note down
You’d add minä mainly for emphasis or contrast (e.g., Minä luen..., mutta sinä et lue...).

Why is uutiskirje in the form uutiskirjettä (partitive) instead of something like uutiskirjeen?

uutiskirjettä is the partitive singular. With many verbs (including lukea = to read), the object can be:

partitive when the action is ongoing, incomplete, or viewed as “some/at it”
total object (often genitive/accusative) when the action is completed

So:

luen uutiskirjettä = I am reading / I read (some of) the newsletter (focus on the process)
luen uutiskirjeen = I will read / I read the whole newsletter (completed)

In this sentence, the “when I’m reading” idea fits the partitive well.

What case is yksi in yhden, and why does it change?

yhden is the genitive form of yksi (one). It changes because it’s part of the object phrase yhden vaikean sanan (one difficult word) and Finnish marks objects with case.
In many common sentences, a singular “countable whole object” appears in a total-object form, which often looks like the genitive in the singular (especially in affirmative clauses).

Why do we have yhden vaikean sanan and not yksi vaikea sana?

yksi vaikea sana is the basic dictionary/nominative phrase (one difficult word) and could be used as a subject (e.g., Yksi vaikea sana on...).
But here it’s the object of merkitsen (I note down), so it takes an object case form:

yhden (genitive form of yksi)
vaikean (genitive form of vaikea)
sanan (genitive-looking total object form of sana)

So the whole object noun phrase is declined to match its role.

Is sanan genitive or accusative? I’ve heard Finnish has a tricky object case.

Form-wise, sanan looks like genitive singular. Function-wise, in an affirmative sentence like this, it’s best to think of it as a total object (often taught as “accusative” in function).
Finnish “accusative” is mostly visible only in:

personal pronouns (e.g., minut, sinut)
the -t form in some contexts (e.g., kirjoita sana vs. kirjoita sana(t) etc. depends on structure)

For most singular nouns, the total object form is identical to the genitive, so learners often just learn the pattern: affirmative + completed/whole object ⇒ genitive-looking singular.

Why is muistikirjaan used, and what does the ending -jaan mean?

muistikirjaan is illative case, meaning into (or “into/onto” in the sense of entering a place/container). Here it’s idiomatic for “write into a notebook”: (into) my notebook.
The illative singular is formed in different ways; for muistikirja it becomes muistikirjaan. The double aa is part of the illative formation for this word type.

Could the sentence be rearranged? How flexible is the word order?

Yes, word order is flexible, but it changes emphasis. These are all possible with slightly different focus:

Kun luen uutiskirjettä, merkitsen yhden vaikean sanan muistikirjaan. (neutral: time clause first)
Merkitsen yhden vaikean sanan muistikirjaan, kun luen uutiskirjettä. (emphasis on the main action first)
Kun luen uutiskirjettä, merkitsen muistikirjaan yhden vaikean sanan. (slightly more focus on where you write it)

Cases carry the grammatical roles, so Finnish can move parts around more freely than English.