Á meðan þvotturinn er í gangi setur hún diskana í uppþvottavélina.

Á meðan is a very common fixed phrase meaning while / as long as when it introduces a full clause with a verb, like Á meðan þvotturinn er í gangi (while the laundry is running).
You can also see meðan on its own in similar meanings, but á meðan is extremely frequent and often feels like the default “while” introducing a time clause.

Yes. Á meðan ... introduces a subordinate (dependent) time clause. Inside it you have normal clause structure:

• þvotturinn = the subject
• er = the verb (is)
• í gangi = a phrase meaning in progress / running

Then the main clause follows: setur hún ...

It separates the subordinate clause from the main clause:
Á meðan ... , setur hún ...
This is very common punctuation in Icelandic when a dependent clause comes first.

Icelandic is generally a V2 (verb-second) language in main clauses. That means: if something else is placed first (here, the whole time clause), the finite verb typically comes next.
So after the fronted time clause, you get:
1) (Fronted element) Á meðan ...
2) Verb setur
3) Subject hún
…then the rest.

þvottur can mean laundry or a wash (cycle) depending on context. Here, with er í gangi (“is running”), it strongly suggests the wash / laundry cycle (or “the laundry” as an ongoing process).
The -urinn is the definite article attached to the noun (Icelandic often adds “the” as an ending).

• þvottur = laundry / a wash
• þvotturinn = the laundry / the wash

í gangi is an idiomatic expression meaning in progress / running / underway.
Literally it’s like “in motion/going,” based on gangur (movement/going), but in everyday Icelandic vera í gangi is the standard way to say something (a machine, event, process) is currently happening or operating.

Because í takes different cases depending on meaning. When í describes a state/location (“in/at” as a situation), it takes the dative.
So:

• gangur (nominative dictionary form)
• gangi (dative singular) after í in í gangi

Because diskana is the direct object of setur (she puts...), and setja (“to put”) takes an object in the accusative case. Also it’s definite (the dishes/plates).
Roughly:

• diskar = dishes/plates (indefinite plural, nominative)
• diskarnir = the dishes/plates (definite plural, nominative; often a subject form)
• diskana = the dishes/plates (definite plural, accusative; fits as the object here)

It’s the classic Icelandic pattern:

• í + dative = location/state (in, no movement) → í gangi (dative)
• í + accusative = movement/direction (into) → í uppþvottavélina (accusative)

So the sentence contrasts nicely: the wash is in progress (state), and she puts the dishes into the dishwasher (movement).

That ending marks definite + the case needed after í with motion (accusative).

• uppþvottavél = a dishwasher
• uppþvottavélin = the dishwasher (nominative)
• uppþvottavélina = the dishwasher (accusative), used here because it’s the destination of motion (into the dishwasher)

It’s a compound of:

• uppþvottur = washing up / dishwashing
• vél = machine
  So uppþvottavél literally means dishwashing machine → dishwasher.

Yes, setur is present tense (3rd person singular) from the verb setja (to put / to set / to place).
Basic pattern:

• að setja = to put
• hún setur = she puts