Learn Hardcore Icelandic: Við vonum að það komi framhald af þessu netnámskeiði næsta ár. - We hope that there will be a continuation of this online course next year.

Using kemur would sound more like you are stating a fact (we hope that a continuation comes / is coming), and is less idiomatic in careful, standard Icelandic. In colloquial speech some people might use the indicative, but komi is the “textbook” choice here.

What exactly does að do here? Is it like English that? Can it be left out?

Here að is a subordinating conjunction (often called a complementizer). It introduces the clause that functions as the object of vonum:

Við vonum [að það komi framhald …]
= We hope [that there will be a continuation …]

Yes, it corresponds closely to English that in this use.

Unlike English, Icelandic normally does not drop this að. In English you can say:

We hope (that) there will be a continuation… (and omit that)

In Icelandic, you should keep að:

Við vonum að það komi framhald… ✔
*Við vonum það komi framhald… ✘ (ungrammatical in standard Icelandic)

What is the function of það in að það komi framhald…? Does it refer to anything?

Here það is a dummy (expletive) subject, similar to English there in sentences like:

There will be a continuation…

It does not refer to any concrete thing (it is not it = some specific object). The real “new information” of the clause is framhald af þessu netnámskeiði.

So structurally:

að það komi framhald af þessu netnámskeiði
≈ that there will come a continuation of this online course

This is the same pattern you see in:

Það kom gestur. – A guest came.
Það verður veisla. – There will be a party.

What does framhald af mean, and is af always used with framhald in this sense?

Framhald means continuation / sequel / follow‑up.

When you say a continuation of X, Icelandic normally uses the pattern:

framhald af + dative

So:

framhald af þessu netnámskeiði
= a continuation of this online course

Other examples:

framhald af sögunni – a continuation of the story
framhald af námskeiðinu – a continuation of the course

Using another preposition, like á, would not be idiomatic in this meaning. So yes, in this “continuation of something” sense, you expect af with the dative.

Why is þessu netnámskeiði in the dative case?

Because the preposition af in this construction governs the dative case.

Preposition: af
Case required: dative

So both the demonstrative and the noun phrase must be in the dative:

þetta netnámskeið – this online course (nominative/accusative)
þessu netnámskeiði – this online course (dative)

Breakdown:

þessu: dative singular, neuter, of þessi
netnámskeiði: dative singular of netnámskeið

Because of af, you cannot use þetta netnámskeið here; it must be þessu netnámskeiði.

What is netnámskeiði exactly? How is this word built and declined?

Netnámskeiði is the dative singular of the noun netnámskeið.

Structure:

net – “net / internet / online” (used as a first element in compounds)
námskeið – “course (of study)”

Together:

netnámskeið – online course

Grammar:

Gender: neuter (like námskeið)
Nominative singular: netnámskeið
Accusative singular: netnámskeið
Dative singular: netnámskeiði (this is the form in the sentence)
Genitive singular: netnámskeiðs

Plural (unchanged stem, typical neuter pattern):

Nom./acc. plural: netnámskeið
Dat. plural: netnámskeiðum
Gen. plural: netnámskeiða

Why is it næsta ár and not some other form like næstu ári?

Næsta ár is a fixed, very common way to say next year.

Points to note:

Ár is a neuter noun.
Here, næsta ár functions as a time expression and is in the accusative case (accusative of time is very common in Icelandic).
Næsta is the weak declension of the adjective næstur (“next”), in accusative singular neuter to agree with ár.

So:

næsta ár – next year (accusative of time)
síðasta ár – last year (same pattern)

A form like næstu ári would be dative, which would suggest a preposition like í (in the next year = í næsta ár / á næsta ári), and it would not match the simple adverbial idiom used here. In this sentence, the natural form is næsta ár.

Can næsta ár appear in a different position in the sentence, or must it be at the end?

Time expressions like næsta ár are fairly flexible in Icelandic word order.

Your sentence:

Við vonum að það komi framhald af þessu netnámskeiði næsta ár.

This is very natural: the time expression comes at the end of the subordinate clause.

Other possibilities (all grammatical, with slightly different emphasis):

Við vonum að næsta ár komi framhald af þessu netnámskeiði.
(Focus a bit more on next year.)
Næsta ár vonum við að það komi framhald af þessu netnámskeiði.
(Puts strong emphasis on next year; “As for next year, we hope…”.)

Inside the að‑clause, the neutral order is usually:

subject (það) – verb (komi) – main content (framhald af þessu netnámskeiði) – time adverbial (næsta ár),

which is exactly what you have. So it does not have to be at the very end of the whole sentence, but its current placement is the most neutral inside that subordinate clause.

Icelandic doesn’t have a separate future tense, so how is the future meaning expressed here? Could we say muni koma instead of komi?

Icelandic typically expresses future meaning in two main ways:

Simple present form (often enough by itself)
Munu + infinitive (a periphrastic “future”)

In your sentence, the future time is already made clear by næsta ár, so the present subjunctive komi naturally gets a future interpretation:

að það komi framhald… næsta ár
= that a continuation will come next year

You could also say:

Við vonum að það muni koma framhald af þessu netnámskeiði næsta ár.

This is grammatically fine, but muni koma tends to sound a bit more like a prediction or a more concrete statement about the future. With a verb of hope, the simple komi is shorter and very idiomatic, and the time phrase næsta ár is enough to show it’s about the future.

Could we say Við vonum að það verði framhald… instead of komi? Is there a difference between komi and verði here?

Yes, you can say:

Við vonum að það verði framhald af þessu netnámskeiði næsta ár.

Here verði is the subjunctive of verða (“become / turn into / happen / be (in future sense)”).

Nuance:

komi framhald
– literally “that a continuation comes”
– evokes the idea that a new instalment or follow‑up will appear or come out.
verði framhald
– literally “that there will be / that there becomes a continuation”
– slightly more neutral, focusing on the existence of a continuation.

Both are natural; verði framhald might actually be more common in some contexts, but komi framhald is also fine and a bit more “event‑like” in flavour.

What is the difference between að vona and að vonast til, and could we say Við vonumst til að það komi framhald…?

Both relate to hoping, but they are used slightly differently.

Að vona – to hope
- Takes a clause directly with að:
  - Við vonum að það komi framhald…
    We hope that there will be a continuation…
Að vonast til (einhvers) – to hope for (something)
- Often followed by genitive or by að‑clause meaning “to hope to / hope that”:
  - Við vonumst til að það komi framhald af þessu netnámskeiði.
    We hope (for it) that there will be a continuation of this online course.

Your alternative:

Við vonumst til að það komi framhald af þessu netnámskeiði næsta ár.

is grammatical and natural. It feels a bit more formal or slightly heavier than Við vonum að…, but both are correct and commonly used.