Læknirinn skrifar lyfseðil, en hún vill lesa hann áður en hún fer í apótekið.

-inn is the suffixed definite article, so læknirinn means the doctor. Icelandic usually attaches the to the end of the noun:

• læknir = (a) doctor
• læknirinn = the doctor

Because lyfseðill (a prescription) is the dictionary form (nominative), but here it’s the direct object of skrifar (writes), so it takes the accusative singular:

• lyfseðill (nom. sg.)
• lyfseðil (acc. sg.)

Skrifa lyfseðil is grammatical and understandable: write a prescription. Very commonly you’ll also hear:

• skrifa út lyfseðil = prescribe / write out a prescription
  Both are used; út just makes it feel more idiomatic/explicit.

The second clause has its own subject: en hún vill... = but she wants...
This hún could refer to: 1) a different person (often the patient), or
2) the same person if the doctor is a woman.

Even if the noun læknirinn is grammatically masculine, Icelandic can still use hún for a female doctor because pronouns can follow the person’s real gender.

en means but and connects two main clauses. Icelandic normally uses a comma before en when it joins independent clauses, just like English often does:

• Læknirinn skrifar lyfseðil, en hún vill...

Because lyfseðill is a masculine noun, and Icelandic pronouns must match the noun’s grammatical gender. So hann here means it (the prescription), not “him” in the human sense.

Because the verb lesa (to read) takes a direct object in the accusative, and hann is accusative masculine singular.

Quick case reminder for this pronoun:

• hann = nominative (he / it)
• hann = accusative (him / it)
• honum = dative (to him / to it)
• hans = genitive (his / its)

(han is not the standard accusative form; you want hann here.)

áður en means before and introduces a subordinate time clause:

• ... áður en hún fer í apótekið = ... before she goes to the pharmacy

In subordinate clauses, Icelandic does not use the main-clause “verb-second” pattern as a rule you have to force; you can generally keep normal subject–verb order like hún fer.

Icelandic often uses the present tense for scheduled/expected future actions, especially in time clauses with words like áður en:

• áður en hún fer... = before she goes... (future meaning)

English does something similar: you say before she goes, not before she will go.

With í:

• í + accusative often indicates movement into/toward a place
• í + dative often indicates location inside a place

So:

• (að) fara í apótekið = to go to/into the pharmacy (motion → accusative)
• vera í apótekinu = to be in the pharmacy (location → dative)

-ið is the definite article for many neuter singular nouns. So:

• apótek = a pharmacy
• apótekið = the pharmacy

Yes. Icelandic normally requires an explicit subject in both clauses. You can’t usually drop hún the way English might sometimes allow in informal style. The repetition is completely natural:

• hún vill lesa hann áður en hún fer...