Learn Hardcore Finnish: Jään kotiin, kunnes netti toimii taas. - I stay at home until the internet works again.

jään = I stay / I’m staying / I will stay (Finnish present often covers near-future too)
jäin = I stayed / I remained (past)
There is no form jäänte; the personal ending for I in the present is built into jään.

What exactly does jäädä mean here—stay or remain?

In this sentence, jäädä kotiin is the idiomatic way to say stay at home (instead of going out / instead of leaving).
More generally:

jäädä can mean stay behind / remain / be left, and with a place in the right case it often means stay (in/at/to that place).

Why is kotiin used instead of kotona or koti?

kotiin is the illative case of koti and means (to) home / into home, i.e. the destination or resulting location. With jäädä, Finnish commonly uses the illative to express staying to/at a place as the outcome: jäädä kotiin = stay home (end up staying at home).

Compare:

Olen kotona. = I am at home. (state/location: inessive/adessive-type meaning)
Menen kotiin. = I go home. (direction: illative)
Jään kotiin. = I’m staying home (instead of leaving). (very common, uses illative)

What does kunnes mean, and how is it different from kun?

kunnes means until (an end point). It sets a limit: you stay home up to the moment when something happens.

kun usually means when (time at which) or sometimes because.

Jään kotiin, kun netti toimii taas. would sound like: I stay home when the internet works again (odd/ambiguous).
Jään kotiin, kunnes netti toimii taas. clearly means: I’ll stay home until the internet works again.

Why is there a comma: Jään kotiin, kunnes …?

Finnish typically uses a comma before a subordinate clause. The part starting with kunnes is a subordinate clause (until …), so the comma is standard:

Main clause: Jään kotiin
Subordinate clause: kunnes netti toimii taas

Why is it netti and not internet or something else?

netti is a very common colloquial/neutral everyday word for the internet in Finnish. You’ll also see:

internet (more international-looking, also used)
Internet (sometimes capitalized, more formal/older style) In speech and casual writing, netti is extremely common.

What form is toimii, and why not toimii taas netti or toimin?

toimii is the 3rd person singular present of toimia (to work / to function):

netti toimii = the internet works / is working

Word order netti toimii is the neutral subject–verb order. You can rearrange for emphasis, but the neutral choice here is netti toimii.

Also, toimin would mean I work / I function, which doesn’t match the subject netti.

Does toimii mean works as in “has a job,” or “functions”?

Here it means functions / works (properly), as in a device/service working:

Netti toimii. = The internet works.
Finnish uses toimia very broadly for functioning/working (internet, phone, machine, plan, method, etc.).

What does taas mean here? Is it the same as “again”?

Yes—here taas means again / once more: the internet wasn’t working, and you’re waiting for it to work again.

taas can also mean on the other hand / in turn in other contexts, but with toimii taas it’s clearly the again meaning.

Is this sentence talking about the future (“I’ll stay home”) or the present (“I’m staying home”)?

It can be either, depending on context. Finnish present tense often covers both:

Jään kotiin can mean I’m staying home (now) or I’ll stay home (decision/plan).
The until clause (kunnes …) makes it sound like a plan/decision lasting up to a point, which English often expresses with I’ll stay home until…, but I’m staying home until… can also fit.

Could I replace kunnes with siihen asti kun or something similar?

Yes. Some common alternatives:

Jään kotiin, kunnes netti toimii taas. (compact, very natural)
Jään kotiin siihen asti, kunnes netti toimii taas. (until that point…, a bit heavier, extra emphasis)
Jään kotiin siihen saakka, kunnes netti toimii taas. (similar, slightly more formal)

In everyday Finnish, kunnes alone is usually the cleanest choice.