Learn Hardcore Icelandic: Ég borða eggið í morgunmat.

egg = base form (nominative/accusative singular, indefinite)
eggið = egg
- -ið, the definite ending for neuter nouns in the singular accusative (and nominative)

So Ég borða eggið is I eat the egg, talking about a specific egg.
If you wanted to say I eat egg(s) for breakfast in a general way, you would more likely say Ég borða egg í morgunmat (no -ið).

What grammatical cases are used in eggið and morgunmat, and why?

eggið is in the accusative case, because it is the direct object of the verb borða (to eat). Most Icelandic verbs take the accusative for their direct object.
morgunmat is in the dative case after the preposition í when it means a location/time “in/at” (rather than movement “into”).

í can take:

accusative = motion into (e.g. í skólann – into the school)
dative = location / time (e.g. í skólanum – in the school)

Here í morgunmat is a kind of “time/location” expression: at breakfast / for breakfast, so it uses the dative.

Why is it í morgunmat and not something like fyrir morgunmat to say “for breakfast”?

In Icelandic, í morgunmat is the natural, idiomatic way to say “for breakfast / at breakfast”. Literally it’s closer to “in breakfast”, but that’s just how the phrase is formed.

í morgunmat = at/for breakfast (what you have as your breakfast)
fyrir morgunmat usually means “before breakfast” (in time), not “for breakfast”

So Ég borða eggið í morgunmat is “I eat the egg as part of breakfast,” not “I eat the egg before breakfast.”

Why doesn’t morgunmat have a definite ending (no -inn or -num)?

Here, morgunmat is used in a generic sense: “for breakfast” in general, not a specific “the breakfast”.

Compare:

í morgunmat – for breakfast (in general)
í morgunmatnum – in the breakfast / during the (specific) breakfast

The second one (í morgunmatnum) can be used if you are talking about one particular breakfast event.
But for a normal, neutral statement about what you eat for breakfast, the bare form í morgunmat is more common.

What is the infinitive of borða, and how is it conjugated here?

The infinitive is að borða — to eat.

In your sentence you have:

ég borða = I eat

Present tense of borða (singular):

ég borða – I eat
þú borðar – you (singular) eat
hann / hún / það borðar – he / she / it eats

So borða here is the 1st person singular present form that matches the subject Ég.

Does Ég borða eggið mean “I eat the egg now” or “I usually eat the egg for breakfast”?

It can mean either, depending on context, just like English “I eat the egg” can be a present or habitual statement, but in isolation it sounds more like a description of what is happening now.

For a habitual meaning, Icelandic often adds a time phrase:

Ég borða egg í morgunmat á morgnana. – I eat egg(s) for breakfast in the mornings.
Ég borða oft egg í morgunmat. – I often eat egg(s) for breakfast.

Your sentence Ég borða eggið í morgunmat most naturally describes a specific egg in a particular breakfast situation.

Can I drop the Ég and just say Borða eggið í morgunmat?

No, not in normal Icelandic. Unlike English imperatives, a plain statement almost always needs an explicit subject pronoun.

Ég borða eggið í morgunmat. – I eat the egg for breakfast.
Borðaðu eggið í morgunmat. – Eat the egg for breakfast. (imperative, different form)

So for a normal declarative sentence in the present, you keep Ég.

Is the word order fixed? Could I say Ég borða í morgunmat eggið instead?

The normal and most natural order here is:

Ég borða eggið í morgunmat.
Subject – Verb – Object – Time/Place

You can move elements around for emphasis, but some orders sound odd or marked for a beginner:

Í morgunmat borða ég eggið. – OK, emphasizes “at breakfast.”
Ég borða í morgunmat eggið. – Grammatically possible, but sounds unnatural/awkward in everyday speech.

As a learner, it’s best to stick to Subject – Verb – Object – Time/Place unless you have a specific reason to change the order.

How would I talk about eggs in the plural instead of “the egg”?

The noun egg is neuter, and its forms look a bit unusual because singular and plural indefinite are identical:

egg = an egg / eggs (indefinite, both singular and plural, nominative/accusative)
eggið = the egg (definite singular)
eggin = the eggs (definite plural)

Examples:

Ég borða egg í morgunmat. – I eat egg / eggs for breakfast. (general, no article)
Ég borða eggin í morgunmat. – I eat the eggs for breakfast. (specific eggs)

How do you pronounce Ég borða eggið í morgunmat?

Approximate pronunciation using English-like syllables:

Ég – like “yeah” with a soft g/“gh” at the end: “yegh”
borða – “BOR-tha” (the ð is like the th in this)
eggið – roughly “EG-ith” (the gg is hard, like “egg”; the ð again like th in this)
í – like English “ee” in see
morgunmat – approximately “MOR-gun-maht” (the t is clear at the end)

Stress is mainly on the first syllable of each word: ÉG BOR-ða EG-gið Í MOR-gun-mat.

What are the genders of egg and morgunmatur, and do they matter here?

egg is neuter. That’s why its definite singular ending is -ið → eggið.
morgunmatur (the full base form) is masculine; its dative singular (without article) is morgunmat.

Gender matters because it determines:

what definite ending you add (-ið for neuter singular here)
how the noun declines in each case (nominative, accusative, dative, genitive)

In your sentence:

neuter object in accusative: eggið
masculine time expression in dative after í: morgunmat