Learn Hardcore Icelandic: Mig langar í mjúka stólinn.

langar is an impersonal verb (“to long for, to feel like”), not a personal 1st-person verb.
It never takes a nominative subject (ég); instead the experiencer goes in the accusative (or sometimes dative), hence mig.
By contrast, vil (“want”) is a regular 1st-person verb and would be used with ég as its subject: Ég vil stólinn (“I want the chair”).

Why is mig (accusative) used instead of ég (nominative) or mér (dative)?

Impersonal verbs like langar put the person feeling the desire into the object case.
You will often see mér langar in more formal or written Icelandic (dative), but colloquially mig langar (accusative) is extremely common.
You cannot use ég here because that’s reserved for true subjects of personal verbs.

When do you use langar í + noun versus langar að + verb?

Use langar í + [accusative noun] to express “I want/have a craving for something.”
e.g. Mig langar í ís (“I feel like (eating) ice cream.”)
Use langar að + [infinitive] to express “I want to do something.”
e.g. Mig langar að borða ís (“I feel like eating ice cream.”)

What case does í govern in this sentence, and why?

Here í is part of the idiom langar í. It always takes the accusative.
In general, í + accusative often signals a dynamic or “toward” sense; with langar it simply frames the thing desired.

Why is stólinn in the definite form (ending in -inn) instead of just stóll?

English “the soft chair” is specific, so Icelandic marks definiteness by adding the suffix -inn for a masculine singular noun:
stóll → stólinn (“the chair”).

Why is the adjective mjúka and not the dictionary form mjúkur?

Adjectives agree in gender, number and case.
After í + accusative, adjectives take the strong declension, even if the noun is definite.
Masculine singular strong accusative of mjúkur is mjúka, so you get mjúka stólinn.

How would you say “I want a soft chair” (an indefinite chair) in Icelandic?

Use the indefinite accusative strong forms:
Mig langar í mjúkan stól.
(Here mjúkan is masculine singular strong accusative of mjúkur, and stól is indefinite.)

Where do you put ekki to make it negative (“I don’t want the soft chair”)?

How would you ask “Do you want the soft chair?” using this structure?

Invert verb and experiencer pronoun, keeping the same cases:
Langar þig í mjúka stólinn?
(Or more formally with dative: Langar þér í mjúka stólinn?)