Learn Hardcore Icelandic: Ég snerti eyrað mitt.

snerti is the past tense (preterite) of snerta (“to touch”), conjugated for 1st person singular.
If you wanted present tense (“I touch”), you’d say ég snertir eyrað mitt.

What case is eyrað in after snerta, and why does it look like nominative?

snerta is a transitive verb that takes a direct object in the accusative case.
eyrað is neuter singular, and in Icelandic neuter nouns have the same form in nominative and accusative.
So even though it’s an accusative object, eyrað looks identical to its nominative form.

Why is the possessive mitt placed after eyrað instead of before it?

When a noun is definite (here because of the -ð ending, “the ear”), the possessive pronoun follows the noun in Icelandic.
Thus eyrað mitt literally means “the ear of mine.” If the noun were indefinite you’d place it before, e.g. mitt eyra (“my ear,” without “the”).

Could I also say Ég snerti mitt eyra? If so, what’s the nuance?

Yes – Ég snerti mitt eyra uses the indefinite form mitt eyra (“my ear”) instead of the definite eyrað mitt (“the ear of mine”).
Both are grammatically correct. The definite version can feel a bit more specific or emphatic (“I touched my own ear”).

Why do we include Ég here? Can you drop it like in Spanish or Norwegian?

Icelandic verbs are conjugated for person, but subject pronouns aren’t always dropped the way they are in fully pro-drop languages.
In colloquial speech you might omit Ég, but in standard Icelandic you usually keep it for clarity and emphasis.

How would you form “I have touched my ear” in Icelandic?

You’d use the perfect with the auxiliary hafa plus the past participle snert:
Ég hef snert eyrað mitt.
Here snert is the supine/past participle of snerta, and hef is 1st person singular present of hafa.