Learn Hardcore Finnish: Ajastin sammuu automaattisesti, kun ruutuaika on täynnä. - The timer turns itself off automatically when the screen time is full.

Ajastin means a timer (a device or function that turns something on/off at a set time).
It is derived from the verb ajastaa (to set a timer, to schedule) plus the device-suffix -in, which often marks tools or instruments (compare lämmitin = heater, from lämmittää).
So ajastin is literally something like a thing for timing.

Why is it sammuu here, and what is the difference between sammua and sammuttaa?

Sammua is an intransitive verb: to go out / to turn off / to be extinguished by itself.
Sammuttaa is the corresponding transitive verb: to turn something off / to extinguish something.

Ajastin sammuu. – The timer turns off (by itself).
Sammutan ajastimen. – I turn off the timer.

In the sentence, sammuu is 3rd person singular (he/she/it), present tense of sammua: the timer goes off / turns off.

How is the form sammuu built from sammua? Why are there two u’s?

The base form is sammua. For the present tense 3rd person singular, Finnish removes the final -a and adds a personal ending.
The stem is sammu-, and the 3rd person singular ending is -u, so:

sammu-
- -u → sammuu

That is why you see a double u at the end. It is regular for type-6 verbs like paeta → pakenee, rohjeta → rohkenee, sammua → sammuu.

What does ruutuaika mean, and why is it one word?

Ruutuaika is a compound noun: ruutu (screen) + aika (time) → screen time.
Finnish very often combines related nouns into a single compound word rather than writing them separately.
So ruutuaika refers to the amount of time spent in front of a screen (phone, tablet, TV, etc.).

Why is ruutuaika in the form ruutuaika, not ruutuajan or ruutuaikaa here?

In kun ruutuaika on täynnä, the word ruutuaika is the subject of the verb on (is), so it appears in the nominative case (dictionary form).
The structure is:

subject: ruutuaika
verb: on
predicative: täynnä

If you changed the role of ruutuaika in the sentence (for example, made it an object or showed possession), then another case like partitive or genitive could appear, but not in this basic subject–verb–complement pattern.

What does täynnä mean here, and why is it on täynnä instead of using an adjective like täysi?

Täynnä means full in the sense of completely filled / used up. With olla (to be), olla täynnä is a very common fixed expression:

Lasi on täynnä. – The glass is full.
Kalenteri on täynnä. – The calendar is full (no free time).

Historically täynnä is in the essive case, but in modern Finnish it behaves almost like a special predicate form that you just memorize.
You could say ruutuaika on täysi, but it sounds less natural and more like complete than (time) used up. For the idea that allowed screen time has been used, on täynnä is the idiomatic choice.

What is the exact role of kun in this sentence? Does it mean when or because?

Here kun is a conjunction meaning when in a temporal sense:

Ajastin sammuu automaattisesti, kun ruutuaika on täynnä.
→ The timer turns off automatically when the screen time is full (used up).

Kun can also sometimes mean because in Finnish, but in this sentence the natural reading is temporal: first the screen time is used up, and at that time the timer turns off.

Why is there a comma before kun?

Finnish normally uses a comma to separate a main clause from a subordinate clause, and a kun-clause is a subordinate clause.
So you put a comma before kun if it follows the main clause:

Ajastin sammuu automaattisesti, kun ruutuaika on täynnä.

If you start with the kun-clause, the comma comes after it:

Kun ruutuaika on täynnä, ajastin sammuu automaattisesti.

Both orders are correct and natural.

The English meaning is future-like: “will turn off”. Why is Finnish using the present tense sammuu and on?

Finnish usually uses the present tense for:

general facts and rules,
scheduled or automatic future events,
instructions and user interface messages.

So Ajastin sammuu automaattisesti can correspond to English The timer turns off automatically or The timer will turn off automatically, depending on context.
You normally do not need a special future tense in Finnish; the present tense plus context covers this meaning.

What kind of word is automaattisesti, and how is it related to automaattinen?

Automaattisesti is an adverb meaning automatically.
It is formed from the adjective automaattinen (automatic) plus the adverb-forming suffix -sti:

automaattinen (automatic) → automaattisesti (automatically)
nopea (fast) → nopeasti (quickly)
selvä (clear) → selvästi (clearly)

So ajastin sammuu automaattisesti literally says the timer goes off in an automatic way, i.e. by itself, without manual action.