Learn Hardcore Finnish: Allekirjoitus puuttuu tästä paperista, joten palaan palvelutiskille. - The signature is missing from this paper, so I return to the service counter.

Because allekirjoitus is the grammatical subject of the clause: Allekirjoitus puuttuu... = The signature is missing...
With puuttua (to be missing/lacking), the thing that is missing is typically the subject in the nominative (or sometimes plural nominative):

Allekirjoitus puuttuu. = The signature is missing.
Allekirjoitukset puuttuvat. = The signatures are missing.

What does the verb puuttuu mean here, and how is it used grammatically?

Puuttuu is the 3rd person singular present of puuttua. In this pattern:

X puuttuu Y:stä
literally X is missing from Y.
So Allekirjoitus puuttuu tästä paperista = A/the signature is missing from this paper.

Why is it tästä paperista and not something like tästä paperista puuttuu allekirjoitus or tämä paperi puuttuu allekirjoitus?

Finnish allows flexible word order, and both of these are possible:

Allekirjoitus puuttuu tästä paperista. (focus on the missing signature)
Tästä paperista puuttuu allekirjoitus. (focus on “from this paper” / contrast with other papers)

But tämä paperi puuttuu allekirjoitus is not correct, because puuttua doesn’t work like “paper lacks signature” in that structure. Instead you’d say:

Tästä paperista puuttuu allekirjoitus. (signature is missing from the paper) or use another verb:
Tässä paperissa ei ole allekirjoitusta. (there isn’t a signature in this paper)

Why is paperista in the -sta/-stä form? What case is that?

Paperista is elative case (means out of / from inside something).
With puuttua, Finnish commonly marks the source/container with elative:

puuttuu tästä paperista = missing from this document (from inside it / from its contents)

It’s a standard government pattern: puuttua + elative.

What is tästä exactly? Is it just this?

Tästä = tämä (this) in the elative case: tästä = from this / out of this.
So:

tämä paperi = this paper (nominative)
tästä paperista = from this paper (elative)

Why is there a comma before joten?

Joten (so/therefore) introduces a new clause. In Finnish, when joten connects two independent clauses, a comma is standard:

Allekirjoitus puuttuu tästä paperista, joten palaan palvelutiskille.
= Clause 1, comma, connector, clause 2.

What does joten mean, and how is it different from niin or sen takia?

Joten is a straightforward logical connector meaning so / therefore (cause → result).
Rough comparisons:

joten = therefore (neutral, written/spoken)
niin can mean so/then but is more conversational and can be less explicitly “logical”
sen takia = because of that (more explicit “due to that” phrasing)

In this sentence, joten neatly links “missing signature” → “I’ll go back to the service desk.”

Why is palaan in that form? What verb is it?

Palaan is 1st person singular present of palata = to return / go back.

minä palaan = I return / I go back

It’s used because the speaker is stating their own next action.

Why is it palvelutiskille and not palvelutiskiin or just palvelutiski?

Palvelutiskille is illative case (movement to/toward a place): -lle = “onto/to (a surface or service point)” in typical usage.
A service desk/counter is treated like a point you go to, so -lle is natural:

palaan palvelutiskille = I’ll go back to the service desk/counter

palvelutiskiin (illative -in) would sound like going into the desk/counter (physically inside), which doesn’t fit. The bare nominative palvelutiski wouldn’t mark direction.

What does the compound palvelutiski literally mean?

It’s a compound noun:

palvelu = service
tiski = counter/desk (especially a counter you stand at)

So palvelutiski = service counter / service desk.

Could you also say menen takaisin palvelutiskille instead of palaan palvelutiskille?

Yes, and it’s very close in meaning:

palaan palvelutiskille = I return/go back to the service desk (compact, natural)
menen takaisin palvelutiskille = I go back to the service desk (more explicit: “go” + “back”)

Palata already contains the “back/return” idea, so it often sounds smoother.

Does Allekirjoitus puuttuu imply a specific signature (“the signature”) or just “a signature”?

Finnish without articles can be ambiguous; context decides.
Allekirjoitus puuttuu can mean:

the signature is missing (a particular expected signature)
a signature is missing (there should be a signature, but there isn’t one)

In real life (forms/documents), it usually implies a specific expected signature.