Learn Hardcore Icelandic: Ég geng beint heim eftir vinnu.

Geng is the 1st person singular present tense of the verb að ganga (to walk / to go on foot).
So Ég geng literally means I walk (or I’m walking, depending on context).

Does að ganga always mean “to walk”, or can it mean “to go” in general?

In Icelandic, að ganga primarily means to walk (go on foot). It can also be used more broadly in certain contexts (like “things are going well”), but in a sentence with a destination like heim, it’s normally understood as walking.

Why is the word order Ég geng beint heim... and not something like Ég beint geng heim...?

Icelandic is typically a V2 (verb-second) language in main clauses: the finite verb usually comes second.
Here:

Ég (subject) = position 1
geng (finite verb) = position 2
Then adverbs and other elements follow: beint heim eftir vinnu.

What is beint here—an adjective or an adverb? Why doesn’t it change form?

Beint is functioning as an adverb meaning straight / directly.
Even though it looks like the neuter form of the adjective beinn (straight), Icelandic often uses that form as an adverb, and it doesn’t agree with anything here.

Why is it heim and not í heim or til heim?

Heim is a special adverb of direction meaning (to) home. It usually appears without a preposition.

Ég fer heim = I go home
Ég geng heim = I walk home

You’d use prepositions in other “home-related” phrases, e.g.:

heima = at home (location, not direction)
heim til mömmu = home to (my) mom’s place

Can I switch beint and heim (e.g., Ég geng heim beint eftir vinnu)?

Sometimes you can reorder adverbs, but it affects what sounds most natural and what gets emphasis.
Ég geng beint heim eftir vinnu is a very natural way to say straight home as a unit (beint heim).
Ég geng heim beint... is possible but tends to sound more like an afterthought emphasis on how you go home.

What does eftir mean here, and are there other meanings?

Here eftir means after in a time sense: after work.
But eftir has other common uses too, for example:

leita eftir einhverju = look/search for something
eftir mér = after me (following me)
eftir bókinni = according to the book (this use takes a different case—see below)

What case is vinnu in after eftir?

With eftir meaning after (time), it generally takes the accusative: eftir vinnu.
However, vinna has the same form vinnu in accusative singular and dative singular, so you can’t tell from the ending alone. The rule is about what eftir requires in this meaning.

Why is there no word for “my” in the sentence—does it still mean “after (my) work”?

Yes. Icelandic often leaves possession implicit when it’s obvious from context.
So eftir vinnu commonly means after work (usually “my work / my workday” unless context suggests otherwise). You can specify:

eftir vinnuna mína = after my work (more explicit, can sound a bit heavier)

Is vinna “work” or “the work”? Why no article?

Vinna here is used in a general sense: work / the workday. Icelandic often uses a bare noun for this kind of routine concept.
If you mean a specific, defined “the work” you might see the definite form:

eftir vinnuna = after the work / after work (often still just “after work” in English)

How would I turn this into a yes/no question?

Two common ways:

Gengur þú beint heim eftir vinnu? (more neutral/formal)
Gengurðu beint heim eftir vinnu? (very common in speech; -ðu is attached þú)

How do I negate the sentence?

Put ekki after the finite verb (very common position):

Ég geng ekki beint heim eftir vinnu. = I don’t walk straight home after work.