Learn Hardcore Icelandic: Sjálfstraustið hjálpar henni að kynna sig án þess að vera of feimin. - The self-confidence helps her introduce herself without being too shy.

In Icelandic, the definite article is usually added as a suffix instead of being a separate word, and -ið is the form used for neuter nominative/accusative singular nouns.

What case and role does Sjálfstraustið have in this sentence?

Sjálfstraustið is in the nominative singular and is the subject of the verb hjálpar.

Sjálfstraustið (subject, nominative)
hjálpar (verb)
henni (indirect object, dative)

So the literal structure is: The self‑confidence helps her …

What form is hjálpar, and why not hjálpa here?

The verb is hjálpa (to help) in the dictionary.

hjálpa – I help (1st person singular present)
hjálpar – he/she/it helps (3rd person singular present)

In the sentence, the subject is Sjálfstraustið (it in English), so Icelandic uses the 3rd person singular present:

Sjálfstraustið hjálpar … = The self‑confidence helps …

Why is it henni and not hún or hana?

Henni is the dative singular form of hún (she).

The verb hjálpa takes the dative case for the person being helped:

að hjálpa einhverjum = to help someone (where einhverjum is dative)

Pronoun forms of hún:

Nominative: hún (she) – used for subjects
Accusative: hana (her, object)
Dative: henni (her, object, after certain verbs/prepositions)
Genitive: hennar (her, of her)

Because hjálpa needs the dative, we must use henni:
Sjálfstraustið hjálpar henni … = The self‑confidence helps her …

What does að kynna sig literally mean, and why sig instead of hana?

Kynna means to introduce, to present.
The phrase að kynna sig literally means to introduce oneself.

kynna einhvern = to introduce someone
kynna sig = to introduce oneself

Sig is the 3rd‑person reflexive pronoun (accusative). It is used when the object refers back to the same person as the subject of the clause:

Hún vill kynna sig. = She wants to introduce herself.

If you said kynna hana, it would mean introduce her (some other woman), not herself.

So:

sig = herself/himself/themselves (reflexive)
hana = her (someone else, not reflexive)

What exactly does kynna mean, and how is it used with different objects?

Kynna is a weak verb meaning roughly to introduce, to present, to acquaint.

Common patterns:

kynna einhvern fyrir einhverjum
to introduce someone to someone
- Hann kynnti mig fyrir foreldrum sínum.
  He introduced me to his parents.
kynna sér eitthvað
to familiarize oneself with something, to study/learn about something
- Hún kynnti sér reglurnar.
  She familiarized herself with the rules.
kynna sig
to introduce oneself
- Hún þarf að kynna sig.
  She needs to introduce herself.

In your sentence it’s this last meaning: að kynna sig = to introduce herself.

What does án þess að mean, and how does it work grammatically?

Án þess að is a very common construction meaning without (doing something).

Breakdown:

án = without (a preposition that takes the genitive case)
þess = of that/it, genitive singular of það
að
- infinitive = introduces a verb in the infinitive (to do form)

So:

án þess að kynna sig
literally: without that to introduce herself
natural English: without introducing herself

This is the standard way in Icelandic to say without doing X:

án þess að borða = without eating
án þess að segja neitt = without saying anything

What is feimin, and why does it look different from the dictionary form?

The adjective is usually listed as feiminn (shy).

Adjectives in Icelandic change form to agree with the gender, number, and case of the noun (or pronoun) they describe.

Indefinite nominative singular forms of feiminn:

Masculine: feiminn
Feminine: feimin
Neuter: feimið

In your sentence, the understood subject of að vera of feimin is hún (she), a feminine subject in the nominative. So the adjective must be:

feimin = feminine, nominative, singular

So:

Hún er feimin. = She is shy.
Hann er feiminn. = He is shy.
Barnið er feimið. = The child is shy.

What does of mean before feimin, and how is it different from very shy?

The little word of before an adjective usually means too in the sense of excessively:

of feimin = too shy (shyer than is desirable/acceptable)
of kaldur = too cold
of dýr = too expensive

It is not the same as English very in most contexts.
For very, Icelandic usually uses mjög:

mjög feimin = very shy (but not necessarily too shy)
mjög dýr = very expensive

So að vera of feimin = to be too shy.

Why is the adjective feimin in the nominative when henni is in the dative?

There are two different roles here:

henni is the indirect object of hjálpar and therefore in the dative.
In the phrase að vera of feimin, the adjective feimin belongs to the subject of vera (to be), which is logically the same person as hún.

With copula verbs like vera (to be), the adjective agrees with the subject in the nominative, not with an object in some other case.

So even though we see henni in the dative earlier, the underlying subject for að vera of feimin is:

(hún) að vera of feimin → adjective in nominative feminine: feimin

Why is the verb in að vera of feimin in the infinitive with að?

In Icelandic, when one verb depends on another (like help someone *to do something, want **to be something), the second verb is usually in the *infinitive with að:

hjálpa henni að kynna sig
help her to introduce herself
án þess að vera of feimin
without being too shy

Other examples:

Hún vill að læra. = She wants to learn.
Hann hætti að reykja. = He stopped smoking.

So að vera is simply the standard infinitive to be used after án þess að.